An anomalous mark made in lieu of an identifying signature isn't useless, and commercial practice is not to verify signatures in advance. It is still useful for proving liability for breach of contract, or for proving that a document such as a payment instrument was authorized.
The Act of Signing, Even If Not Identifiable, Has Legal Significance
Signing in any form signifies assent and actual assent is an element of a breach-of-contract claim. Indeed, when literacy was less common, signing with a mere "X" (which is essentially equivalent to what you describe in your fact pattern) was a common way to secure the assent of an illiterate person to a written document, even though it had no association with the person except that it was made in a signature block or on a contract containing the name of the party to be bound.
This is the act that creates liability. Everything else in a breach of contract case or case denying authorization was given for a payment, is about proving that this act happened. The mark of assent does not have to be identifying to be a valid signature. The role of a signature in identifying who signed the document is optional.
In the case of an agreement subject to the statute of frauds, an oral agreement cannot create liability, but a piece of paper with an "X" on it from the party to be charged can create liability, as a matter of substantive law as opposed to as a matter of evidence. And many important contracts are subject to the statute of frauds. (Part performance will usually excuse a failure to have the agreement in writing under the statute of frauds.)
A Signature Distinguishes Drafts and Offers From Authorized Documents
Any kind of mark on the document means that someone signified their assent to the document. The fact that there is a signature from someone distinguishes between mere offers to enter into a contract or authorize a payment or a draft agreement from agreed-to or authorized acts.
Similarly, it isn't uncommon to have someone initial every page of a contract or key terms for the same reason: to show that this is the version of the agreement that was actually agreed to and not some other draft or version that was slipped in after the fact by the party who is suing.
Any Signature is an Implied Warranty of Authority to Sign
If the contract names the parties on its face or in the signature block, any sort of mark on that document is an implied warranty that the person signing is authorized to sign on behalf of the person named.
Even if the signer was an agent of the person who was supposed to sign, this would be some evidence that the person who signed it had "implied authority" to act on behalf of the named person (even if, in fact, the agent was told not to do so), which could bind the person named, even if the person named didn't actually sign it.
There is always some other evidence of the source of the signature
Legal documents and documents pertaining to real estate and testamentary documents would usually have a notarization that would confirm the identity of the signer.
Insurance payout applications in large amounts require a private sector equivalent to a notarization called a "guaranteed signature" that is run by banks, with the guaranteeing bank subject to liability in the event of a claim of fraud.
If an illiterate person signs a document with a mark that is not identifying, often it will be witnessed by someone else (although this is not legally required).
Many firms that accept personal checks as payment also require driver's license or other identification.
Credit card and debit card payments usually require presentation of a physical card as well as a signature. This greatly narrows the range of who could have used the card and also puts an onus on the user that he or she was negligent in allowing someone else to gain access to the card, or in failing to report the card as stolen.
Debit card and ATM transactions usually require entry of a PIN (and ATM transactions and many merchant transactions are videotaped). Use of a PIN shows authorization.
Often people entering into a contract are known to each other and can testify from personal knowledge that the correct person was signing the document.
Many contracts are signed by more than one person (e.g. as tenants or promissory note obligors), and testimony from co-signers can establish that the person with the anomalous signature also signed it.
Contracts transmitted and received via email or fax have the means of transmission as additional evidence of the source of the signed document.
At a minimum there would be some evidence that a contract was submitted to someone by mail or handed to someone, and there would be some evidence of how it was received back (in person, by mail, by courier, etc.). Generally, there would also be a course of dealings that went into negotiating the contract that would narrow the list of suspects.
A court can review other signatures made by the person on documents and if that person routinely uses anomalous signatures, the discrepancy in the signature offered will be given less weight.
The person who allegedly signed the document can be compelled to testify under oath regarding whether or not they signed it. So, in order to deny liability you have to commit a felony (perjury) and if your lawyer knows that you are lying, your lawyer is forbidden from offering up your testimony to a court (at risk of disbarment).
Even an anomalous signature, when taken together with corroborating evidence of this kind, can be powerful evidence of an agreement to a contract or authorization of a payment.
Don't forget that sworn testimony is a form of proof potentially equal to or overriding a written document. It is a common fallacy to say that you have no proof of something simply because it is not supported by a written document.
Also, don't forget that in breach of contract actions, the burden of proof is generally a preponderance of the evidence. The claiming party only has to show that it is more likely than not that the contract was agreed to by the parties in light of the totality of the evidence. This is not a terribly burdensome burden of proof.
Even an anomalous signature means that a defendant has to come up with a theory about how the signature got there that is more plausible than the theory that the defendant signed it in an anomalous manner. Finding a motive for someone else to have signed it is often difficult or impossible.
Keep in mind that most contracts and payment authorizations made without much corroboration are modest in amount. A certain amount of fraud can be absorbed as a cost of doing business. The authorization system has to be good enough to discourage widespread fraud to be economically sufficient, it doesn't have to be absolutely foolproof in the face of people willing to lie to a court and to carefully arrange circumstances so that there can be no corroboration of a signature.
Receipt of Benefit Can Prove Assent to a Contract or At Least Liability
Often contracts are sued on at a time when they aren't completely executory. Someone has performed their responsibilities under the contract. If the person with an anomalous signature received benefit without objection per the terms of the contract under circumstances indicating an intent on the performing party to receive something in exchange, the person with the anomalous signature can be sued for unjust enrichment even if the contract was never signed at all.
For example, suppose someone does forge you name on a promissory note, but the money is deposited in your bank account. You are still obligated to repay the amount deposited with statutory interest, even if you didn't sign the promissory note, in the absence of evidence showing that the money that was deposited was intended as a gift.
The remedies for unjust enrichment may be slightly different than the remedies for breach of contract (e.g., default interest rates, liquidated damages, and attorneys' fees clauses may not be available), but the availability of this partial recovery makes the risk of criminal liability for fraud far less worth it to someone employing this tactic to try to defeat contract liability.
Also receipt of benefit without objection tends to make the allegation that the signature was forged much less plausible, so it favors a finding of actual breach of contract and a conclusion by the court that the signature was not forged.
A variant of this theory is that acceptance of the benefits of a contract which was proposed and of which the person was aware (which can be shown without validating the signature) can be considered an acceptance of the offer in the contract by action, or as a ratification of the contract.