The problem here is that misapplied empathy can lead to terrible decisions. Having Google change their 2FA system for this group would be one such decision. It's similar to the 'think of the kids + terrorism' attacks on encryption. It's socially difficult to argue against these ideas because you are then labeled as a terrible and non-empathetic person, but the solutions themselves make one other thing worse without really being helpful other than for garnering retweets and likes.
In this case, we actually aren't being ambitious enough. Why are we having a system where we give out phones every 12 weeks to each homeless person? We'd probably save money for the program by developing some sort of dedicated device designed to be harder to steal or lose. Maybe a high-autonomy low-powered KaiOS smartphone that can be attached as a strap? It's not like the current devices are working.
Why is it such a hassle to keep the same number after a theft? We could investigate there too. Improving this would be better than decreasing the effectiveness of gmail's measures.
Heck, if we want to focus on Gmail, why not focus on why it's the default choice for the homeless to begin with, as opposed to removing features.
We could try to solve the problem structurally but we prefer the caseworker approach, because it's more easily packaged 'empathy' than actually fixing the homelessness issue. It's like people who travel to developing countries to 'help', when the locals need investments and training facilities, not extra warm bodies. Actually giving homes to the homeless would probably be cheaper than whatever we are doing now, even taking into account the mental illness and drug-abuse problems that factor into this.