Wife And I Own A Home, My Mom (tenant) Wants To Leave It To All 3 Kids

Location: Connecticut. My elderly mother declared bankruptcy years ago and needed a place to live. My wife and I have great credit and good jobs, but live in NY and pay rent so couldn't feasibly afford mortgage payments. We got a mortgage in our name and bought a house - our names on the deed, mom pays the mortgage. This worked well for years, but recently mom referenced leaving the house to my siblings and me equally.

I assume paying the mortgage alone doesn't afford her legal ability to will the house or dictate what will be done with it. Neither of my siblings contributed financially, signed anything, or assume any risk. My wife and I have also paid for new appliances, improvements, etc if that matters.

I lack the technical language, but I guess my mother views the house as "her estate", and that NOT splitting it equally amongst her children would be snubbing my siblings. She recently said "Well we really only used your credit score, I'm the one paying the mortgage." In my view ... no credit, no house.

Furthermore, mom needs a car and wants to take out a home loan. Same deal; we secure a loan with our credit/income, she pays the interest, principle payment comes out of the sale of the house when she dies (assuming a long draw period). My wife and I are uncomfortable assuming another financial risk - and feel a bit used - given the house situation. This disagreement has stalled a larger conversation about estate planning.

Can someone please provide an educated summary to bring the conversation out of emotional-obligation-land and into legal reality? Does my mother have legal standing to influence what happens to the house? Are there technical terms we should be using to clarify our position vs hers when we discuss further? Is there anything we can do to strengthen our claim, or are we already the incontestable legal owners? Also (if the house were split), wouldn't my wife be entitled to an equal share, as we're 50% co-owners on the deed? Any help is deeply appreciated. Thank you!

FWIW, we're amenable to sharing the proceeds with my siblings, to the extent they help support mom. Not amenable to gifting them 2/3 the value of a house we own.

TLDR: Bought a house for mom, my & wife's name on deed and mortgage. Mom pays the mortgage, wants to leave house to me and uninvolved siblings, equal split. Her decision or mine? Thanks!

***EDIT & update**\*: Thank you all for the responses! To address questions:

1. Down payment: Mom was bankrupt and did not make it, a close friend supplied it. I said "lent" and "repaid" above but now I'm not sure it wasn't a gift.

2. The original agreement: There wasn't one, save "it'll appreciate and you'll sell it". Her lease on a rental was ending and this was done FAST. Wife & I never owned a house/mortgage before, mom basically said, "let me do the talking". We didn't know which questions to ask because we were (are) dumb, inexperienced kids and thought we were saving the day.

3. Mortgage fraud: Didn't know this was a thing. I do know ignorantia juris non excusat. Is it potentially fraudulent, hinging on how the mortgage was structured? Or definitely, by virtue of ... using one party's credit and another to pay the mortgage? Not sure if that conversation can continue here or needs a new topic, or is even allowed by the rules. Obviously I want to right anything that's wrong.

4. Can we afford the mortgage payments ourselves? Yes - not comfortably.

5. The car loan: my mother doesn't ask my siblings for money. Hasn't asked the most affluent child for anything, there's history there. I'm trying to honor her wish, but now I'm worried my financial assistance is "our little secret" so she can have unburdened relationships with them.

6. Greed, moral failing and sibling treachery: I appreciate your frankness. I have many blindspots in this situation. Being called out and confronted with so much I didn't see is helpful. I wanted to establish a clear, factual, legal understanding as a baseline from which to make informed decisions, and not rely on a rotation of ignorance, emotion and family drama as my only guide. You've all helped with that.