They clearly haven't abandoned them as that applies generally to property; if you are in fact charging rental fees, you haven't intentionally relinquished the right to control something, as you are exercising that right by charging others for the right to temporarily exercise some portion of it.
There may be rules regarding public rights of way which impact this, though the US Ninth Circuit Court of Appeals has already rejected the claim that property left unattended on a public sidewalk is therefore abandoned and subject to deprivation when the City of Los Angeles used that as an excuse to take the goods belonging to the homeless without due process, so I think what is really going on in those DAs offices is people have made the decision “we don't like what Bird is doing so we aren't going to enforce the law when people commit crimes against them.”
In contrast, the homeless person has not left there property around for the express purpose of letting others use it.
Intending to rent it to others for use, yes.
> If they are not locally permitted to do so, then this act constitutes abandonment
No, it doesn't. It may constitute violation of whatever ordinance does not permit it, but the Lavan v. LA case is directly on point, that whatever regulatory powers local jurisdiction has, a property owner not intending to entirely relinquish control has not abandoned (and the state cannot without violating the due process clause of the 14th amendment treat them as having abandoned) property merely because it is unattended on a public way in violation of some local control. (They may have the right to take the property with due process from the person who remains, until that process has been given, the owner, but that's a remedy against the owner, not abandonment by the owner.)
While I concede that your position is correct academically, I've successfully argued my position in court several times (pro bono, since I only take on criminal def when I believe in the case), so I'm confident that my position is correct where it matters.
And if you're ever actually practiced in the California court system, you would know that you only need to convince a single appellate judge statewide to make valuable precedent. (Because in California, a lower court can follow the precedent of any higher California appellate ruling.)
In California, if you find it and choose to take charge of it, you become a depositary for the legal owner. Additionally, if it is worth $100+, and you can't locate and return it to the owner within a reasonable period, you are required to turn it into the police or sheriff depending on the jurisdiction in which it was found. It may become yours if the owner doesn't claim it from the police/sheriff within 90 days, with some additional requirements if it is worth more than $250.
If it was intentionally abandoned by the owner this doesn't apply.
EDIT: Source, Civil Code § 2080 et seq..