The general rule is that a warrant is required to enter private property (absent constitutional case law exceptions to the warrant requirement such as exigent circumstances and consent), and that a warrant is available only when there is probable cause that a crime has been committed.
Whether the neglect or abuse of an animal constitutes a crime within the meaning of this 4th Amendment requirement could potentially be seen as a gray area, since historically, in the absence of statutory authority in early common law, an owner of an animal had absolute authority to deal with his or her property (the animal) as the owner of the animal saw fit.
The purpose of the statute is to clarify that this conduct by an animal owner constitutes a crime for 4th Amendment search and seizure purposes by making a state law determination that it is a crime, which states can do, even though they can't change the constitutional requirement under the 4th Amendment.
Also, just because a state can authorize law enforcement to get a warrant for any search authorized by the U.S. Constitution, that doesn't mean it has to allow law enforcement to do so in every case where it is constitutional for the state to do so.
The duty to get a warrant for law enforcement to enter onto private property at all arises not only from other state statutes, but also from the 4th Amendment to the U.S. Constitution (as incorporated to apply against state and local governments though the due process clause of the 14th Amendment to the U.S. Constitution).
But, the constitutional requirement has case law exceptions, so it isn't required in all circumstances. In particular, exigent circumstances, and the consent to entry exceptions, which are allowed by constitutional criminal procedure case law, could apply to the requirement to get a warrant in the first place.
But, law enforcement needs to have the authority to search at all with a warrant under state law, for an exception to the warrant requirement to be relevant. This statute appears to carry out that purpose by authorizing searches for this particular purpose. For what it is worth, it is not the best drafted possible statute to achieve this objective, and it could have been written to be more clear, but it still gets the job done.
So, in answer to the top-line question, no, I wouldn't read this statute as requiring a warrant in every possible circumstance in order to go onto private property to check on an animal, although a warrant would be required in every case where an exception to the warrant requirement under 4th Amendment case law does not apply.
Section 578. Is an animal related statute rather than people.
The people involved are the property owners. The property owner's rights in their real property are potentially infringed if there is a warrantless entry.
The human beings owning the animals are potentially violating a law which the State of Missouri wants law enforcement officers to be able to enforce (the relevant laws are the state animal cruelty and agricultural laws expressly referenced in the statute, so, it is irrelevant that "Barry County Missouri has no animal control laws or leash laws").
Among other things these statutes make it a crime if a person "Has custody or ownership of an animal and fails to provide adequate care[.]" As the question claims that: "The definition "Adequate care" is vague as well." But the question also notes that: "The 578 statute has been challenged for being unconstitutionally vague and arbitrary which was overruled but that was serious abuse case."
The state has a right to decide what is and is not illegal. It is not prohibited from banning treatment of animals that is not serious abuse. The state has every right to make it a crime to fail to provide adequate care for an animal, even if that failure to provide adequate car does not constitute severe abuse.
Also, keep in mind that a lawful search requires only probable cause to believe that a crime was committed and a good faith belief that an exception to the warrant requirement is present.
If the law enforcement officer has a good faith belief that the animal will die or seriously suffer or be hidden by the owner in the time that the law enforcement officer reasonably thinks that it will take to get a warrant, the exigent circumstances exception to the warrant requirement applies.
The fact that the lawful search later reveals that a crime was no committed does not mean that the search was improper. A mere belief that an animal was abused or neglected and that exigent circumstance were present with a reasonable factual basis (e.g. a tip from a neighbor who seems credible and claims to have personal knowledge of the facts) will usually suffice to establish probable cause.
So warrant needed or not? and if so, what legal action can be taken
for trespass, rights violations under color of law etc. if any?
If there is a search without a warrant or probable cause was not present, and an exception to the warrant requirement does not apply, and the property owner believes that their 4th Amendment rights were intentionally violated by law enforcement in the warrantless search in violation of clearly established law to the contrary, a civil lawsuit against the law enforcement officer under 42 U.S.C. § 1983 can be brought in state or federal court.
The employer of the law enforcement officer can be sued as well, under the same statute, if the warrantless search in violation of the clearly established constitutional right was made pursuant to an express policy of the law enforcement officer's employer. But the fact that the law enforcement officer violated someone's rights does not automatically make the law enforcement officer's employer civilly liable for the wrong.
In most U.S. states, law enforcement officers are protected by state law governmental immunity from common law trespass lawsuits for their conduct while carrying out their official duties, but I haven't checked specifically to see if that is the case in Missouri.
A claim of a 4th Amendment violation can also be a ground for suppressing evidence obtained with an unlawful search when defending a prosecution under some ordinance or statute that relies upon that evidence.