Errors in the Question
Of course, this is not actually what really happens. What really happens is that the original crime is not tried. It is simply ASSUMED to have taken place and there is a presumption that the original actor was guilty of some crime. Obviously this is incorrect and unjust.
This is incorrect. To convict a person of being an accomplice, in either US or UK courts, there must be evidence proving that the crime took place. This an essential element of the crime. It need not be proved who committed the crime as a principal, although there is often evidence about that. But an essential element of the crime of being an accomplice is that the underlying offense is a crime and did occur. Like all other elements of a crime, this must be proved beyond a reasonable doubt.
Therefore, it would seem that from a theoretical point of view, an accomplice should only be charged and accused once someone else has first been convicted of a crime. Only then should the accomplice be accused of helping that person.
That is not the theory that the legislatures of either the US or the UK have adopted. Indeed I do not think that any common-law country has adopted this theory. The law could be changed, and one might argue that it should be. But what the law should be, as opposed to what it is, is not generally on-topic here, as opposed to in politics.SE.
Accomplice
The LII Definition of "Accomplice" is:
A person who knowingly, voluntarily, or intentionally gives assistance to another in (or in some cases fails to prevent another from) the commission of a crime. An accomplice is criminally liable to the same extent as the principal. An accomplice, unlike an accessory, is typically present when the crime is committed.
Nothing there suggests that the principal must be convicted first.
The Wikipedia article "Accomplice" says:
An accomplice differs from an accessory in that an accomplice is present at the actual crime, and could be prosecuted even if the main criminal (the principal) is not charged or convicted. An accessory is generally not present at the actual crime, and may be subject to lesser penalties than an accomplice or principal.
...
The fairness of the doctrine that the accomplice is still guilty has been subject to much discussion, particularly in cases of capital crimes. Accomplices have been prosecuted for felony murder even if the actual person who committed the murder died at the crime scene or otherwise did not face capital punishment.
In jurisdictions based on the common law, the concept of an accomplice has often been heavily modified by statute, or replaced by new concepts entirely.
In The Wikipedia article "Accessory (legal term) In the section on "England and Wales it is said:
A mens rea {guilty state of mind] is required even when it is not required for the principal offender (for example, when the principal commits a strict liability offence). The defendant must intend to do the acts which he knows will assist or encourage the principal to commit a crime of a certain type. In R v Bainbridge (1960) 1 QB 129 the defendant supplied cutting equipment not knowing exactly what crime was going to be committed, but was convicted because the equipment supplied was not used in the ordinary way, but for a criminal purpose instead. The accomplice must also know of all the essential matters that make the act a crime ...
In the section "United States" it is said that:
U.S. jurisdictions (that is, the federal government and the various state governments) have come to treat accessories before the fact differently from accessories after the fact. All U.S. jurisdictions have effectively eliminated the distinction between accessories before the fact and principals, either by doing away with the category of "accessory before the fact" entirely or by providing that accessories before the fact are guilty of the same offense as principals. The Model Penal Code's definition of accomplice liability includes those who at common law were called accessories before the fact; under the Model Penal Code, accomplices face the same liability as principals. It is now possible to be convicted as an accessory before the fact even though the principal has not been convicted or (in most jurisdictions) even if the principal was acquitted at an earlier trial.(Wayne LaFave, Substantive Criminal Law § 13.1(e) (2d ed. 2003).)
18 U.S. Code § 2 - Principals provides that:
(a) Whoever commits an offense against the United States or aids, abets, counsels, commands, induces or procures its commission, is punishable as a principal.
(b) Whoever willfully causes an act to be done which if directly performed by him or another would be an offense against the United States, is punishable as a principal.
Note that there is no requirement that a principal be first tried or even identified to convict one who abetted a crime. All accomplices are abettors, although not all abettors are accomplices.
The Model Penal Code (MPC) published by the American Law Institute says, in section 2.06 of the code, that:
§ 2.06. Liability for Conduct of Another; Complicity.
(1) A person is guilty of an offense if it is committed by his own conduct or by the conduct of another person for
which he is legally accountable, or both.
(2) A person is legally accountable for the conduct of another person when:
(a) acting with the kind of culpability that is sufficient for the commission of the offense, he causes an innocent or irresponsible person to engage in such conduct; or
(b) he is made accountable for the conduct of such other person by the Code or by the law defining the offense; or
(c) he is an accomplice of such other person in the commission of the offense.
(3) A person is an accomplice of another person in the commission of an offense if:
(a) with the purpose of promoting or facilitating the commission of the offense, he
(i) solicits such other person to commit it, or
(ii) aids or agrees or attempts to aid such other person in planning or committing it, or
(iii) having a legal duty to prevent the commission of the offense, fails to make proper effort so to do; or
(b) his conduct is expressly declared by law to establish his complicity.
(4) When causing a particular result is an element of an offense, an accomplice in the conduct causing such result is an accomplice in the commission of that offense if he acts with the kind of culpability, if any, with respect to that result that is sufficient for the commission of the offense.
...
(6) Unless otherwise provided by the Code or by the law defining the offense, a person is not an accomplice in an offense committed by another person if:
(a) he is a victim of that offense; or
(b) the offense is so defined that his conduct is inevitably incident to its commission; or
(c) he terminates his complicity prior to the commission of the offense and
(i) wholly deprives it of effectiveness in the commission of the offense; or
(ii) gives timely warning to the law enforcement authorities or otherwise makes proper effort to prevent
the commission of the offense.
(7) An accomplice may be convicted on proof of the commission of the offense and of his complicity therein, though the person claimed to have committed the offense has not been prosecuted or convicted or has been convicted of a different offense or degree of offense or has an immunity to prosecution or conviction or has been acquitted. (Emphasis added)
The official commentary on subsection (7) says:
Subsection (7) speaks to the relation between the prosecution of the accomplice and the treatment of the person who is alleged to have committed the offense. In accordance with modern developments, this subsection provides that the accomplice can be prosecuted even though the other person has not been prosecuted or convicted, has been
convicted of a different crime or degree of crime, has an immunity to prosecution or conviction, or has been acquitted.
The MPC was the result of a 10-year review of the laws then existing in the various US states. It was an attempt to rationalize existing laws and provide a consistent framework for criminal law in the US. It was offered to the states for adoption in whole or in part, and several states have adopted significant sections of it, and others have been guided by it in revising their legal codes.
Section 2.06 (7) makes it clear that no previous prosecution of a principal is required for prosecution of an accomplice in any state that has adopted the MPC.
Effect of the Theory of the Question
The question asserts:
Therefore, it would seem that from a theoretical point of view, an accomplice should only be charged and accused once someone else has first been convicted of a crime. Only then should the accomplice be accused of helping that person.
If courts adopted that as a rule, it would mean that if the principal escaped, or died, no accomplice could be prosecuted.
Indeed consider the case of a crime boss, who is often an accomplice to the crimes of his (or her) henchmen. Such a boss would need only to have the actual criminal sent out of the country, or killed, to be quite safe from prosecution under the proposed rule. Hardly an improvement to my eyes.
Such a rule would also mean that an accomplice could not be pressured, by means of a threat of prosecution, to disclose the actual criminal, or testify against that criminal. Again, not desirable.
Use of UK Law
I would add that in my answer to the linked question, sources from several US states were cited, and no UK sources. Indeed UK law does not seem to have been cited in the comments either, but this is a matter on which US and UK law are quite similar.
