I don't think thats how it works and heat pumps aren't that much more expensive than other new heating systems, and over long time they are cheaper even.

How about we tax billionaires enough so that the government can train thousands of people to be heat pump technicians and potentially own their own trades business, and then also pay to buy and install them?

The answer:

Everyone would have heat pumps.

This is a genuine economics question, but would these kinds of things ever even work? In the sense that, billionaires hold a lot of money, yes, but they never use a vast majority of it. That effectively means that money doesn't exist. Just pumping money into something doesn't create people and resources out of which you can create products.

If we were to redistribute wealth equally, how much would that actually help people (other than land and housing since that would definitely help enormously)? Sure some of the production capacity would stop going to producing some of the extremely expensive and resource intensive products such as yachts and the like, but it's not like rich people are buying 100s of ACs just for themselves. Shifting all of that production capacity to other goods I don't feel like is going to lead to that many more consumer products for regular people.

My point (and my actual question/thought) is that it would definitely help a lot of people a lot, but I feel like by how much is overstated, and I feel like the percentage of wealth that the 1% holds doesn't really matter as much as how the wealth in the rest of society is distributed (i.e. if the 1% were twice as rich but the other 99% had their wealth equally distributed, then that would almost be the same as if all of society had its wealth distributed equally)

Edit: Just to clarify, I'm not focusing on the part about whether it's okay or moral for there to be people that are so much richer, I'm just talking about the practical consequences.

It's expensive to get them installed. Instead of ducts running all over your house, the refrigerant line has to run from the outside section to the inside section. It costs as much to get one of these as a normal central air system because it's also complicated. Heat pumps are more efficient in certain situations so that's the tradeoff.