When’s the right time?

Yoy are stuck with caring for her for the next 5 years if you want the taxpayers to pick up the bill and you keep the house. You are right, Medicaid won't turn a blind eye to a large asset transfer 6 months prior to applying, and they shouldn't.

I don't think 5 years of your life is worth a crummy piece of property but, some do.

Unless your Mom is independently wealthy enough to afford the assisted living which would be memory care needs and upward of 10,000 a month, she would be on medical. Keep careful track of her finances if you are serving as her POA and be ready to show 5 years (2 1/2 in California) years worth of records of her finances. When she has spent down in nursing home she would move onto medicaid and what assets she left in terms of a home would be recovered from medicaid on her death, if such assets exist. The first thing you should do is invest in the 350.00 an hour for an hour of elder care time to discuss her finances, her diagnosis and her assets, as well as whether you are currently her conservator or guardian, or wish to be, or if you would rather the State takes over this onerous and burdensome task in terms of her future needs (which would not allow you any input into when/where/how placed, but would be the responsibility of the Fiduciary appointed by the court).
To be frank the time is now or some time ago for me, proving that this is an individual question for an individual caretaker, YOU. Only you can make the decision. I don't know the prognosis on your Mom's diagnosis, but she can live decades more in your good care. Only you can decide when you are ready for this. There is no reason to feel guilt; what you will feel is the other G word which is grief, that this must be the ongoing long ending for your Mom. I am so sorry.

I'm so sorry that you're being worn down by the caregiving. Has your mom ever been diagnosed by a doctor? Because what you describe (her age) sounds like ALZ. This only matters because the behaviors and trajectory are different than other types of dementia.

Has medication for her depression ever been discussed with her doctor? Meds often are very helpful to both the person suffering and the caregiver.

When you say you've "done a poa for her" did she actually sign the paperwork and understanding what she was signing, in front of witnesses and notary? If not, I'm not sure what you have going is legal. This matters most if you have siblings or other family members who may question your authority.

Does NYS's Medicaid pay for any of AL? I'm asking because many states do not, they only cover LTC and MC). There is an Elderly Waiver federal program you can research that provides resources for those who income qualify.

I would start by getting a formal diagnosis if you haven't already done this. Then contact social services for your county to get an in-home assessment for services. Your mom may be beyond AL at this point, requiring MC instead but she needs to be assessed.

As her DPoA I'm not sure you can transfer the title to her home to yourself, but I may be wrong. This may disqualify or delay her for Medicaid. Locate a Medicaid Planner in your area and invest in a consult so that you don't jump through a bunch of hoops only to have it be for naught. This is what I would do. In the meantime see if you can address her depression with some meds. Also, I highly recommend learning about ALZ and dementia. I learned a lot from Teepa Snow videos on YouTube, what dementia is, how it affects our LOs and why, and how to better engage with them so that your caregiving is as least-stressful as possible. You have a family and you and they have priority over your mom. If you burn out then who will be there to help your mom? Blessings to you for making such a sacrifice for your mom to this point.

Assisted Living costs would be out of pocket expense. Can she afford that? (you should not have to pay for AL for her)
the ideal scenario would be that the house would have been sold and the proceeds used to pay for her care. That would be either caregivers that come in or Assisted Living and an eventual transition to Memory Care. Keep in mind that in AL a person can leave anytime they wish. All they have to do is sign out. So if there is a concern mom would wander off AL would not be a good option.
Very possible that mom also has depression, has she seen a doctor? You can not "cure" a hoarder by cleaning the house. Hoarding is the result/symptom of a mental illness that should be treated.

There will be others well versed in the ins and outs of Medicaid that will chime in on the transfer of the property.