1. Everyone should have access to banking, and a $5k nominal/year in most of the G7 economies[0] is small enough it shouldn't attract attention.
2. If you don't require proof of ID, you can't stop people signing up for as many small accounts as they want to.
3. Homeless people may not have any formal proof of ID (either because it's too expensive, or as a direct consequence of why they're homeless in the first place[1]).
You can try to get around #3 with biometrics. But even with AI assistance, the goal here is to prove a person isn't already on the system — which is much harder than merely showing they're the same person as on the ID card they have with them, for reasons which are basically a generalisation of the Birthday problem[2].
[0] one of the G7 is the EU, the worst of the EU is Bulgaria at €12400 GDP (nominal) per capita, Wikipedia reports that "More than a fifth of the labour force work for a minimum wage of $1.16 per hour.[203]" but that citation (which follows) is from 2012 and this is where I stopped going down the rabbit hole: https://web.archive.org/web/20121224023912/http://epp.eurost...
[1] off the top of my head: because they ran away from home as a minor and didn't take sufficient documentation with them; or because they had an untreated mental health problem; or because they're an unregistered migrant; or because they were born in the country but out of the system.