Mustache Anxiety, you are confusing Medicare and Medicaid. Not the same animals by a long shot.
True. But Medicare does not cover nursing homes at all. It will cover hospital stays and rehab and short-term assistance to help you manage, treat, and recover from an illness or injury, but it will not cover long-term care in an assisted living facility. So everything that Mustache Anxiety wrote is entirely accurate about what happens when you get to the point of needing nursing home care -- you pay for your own private facility yourself, and then when your money runs out, you qualify for Medicaid and need to go to whatever facility will provide you a Medicaid bed. And Medicaid will absolutely claw back any assets that were transferred to others within that 5-year lookback period.
So if the primary goal is to transfer those assets to you and not having to spend them down on nursing care, the best approach is definitely to have him transfer his assets to you quickly, given the 5-year lookback.* And then you can leave it untouched for the next five years, at a minimum (because you don't want to have to turn over money that you've already spent). Personally, I'd mentally earmark the money as a "dad care" fund -- even if he says he doesn't want the money to go to his own future care, it is good to have those options when the time comes. So if it turns out that he does need assistance at some point, if you have the money, you can then choose to spend it yourself to provide those things to him.**
*Yes, of course he could choose to blow the money on parties and trips and so on, and Medicaid won't come after him for that. But the stated goal was to pass down the money to his family, not spend it all.
**This is sort of what my dad and uncle have done for my Granny. They bought her a condo, and she insists on paying them rent. And, you know, you don't mess with Granny; if she says that's how it's going to be, that's how it's going to be. So they take the rent checks and put them in a separate fund that they use for condo repairs, medical care, and any other stuff that she needs. She is of course sharp enough to have figured out their workaround, but it allows everyone to save face.