I make yogurt without a yogurt-maker like this quite often. I used the heating pad method. I pour the mixture into a quart glass jar, place it on the heating pad, wrap it in a towel, and put a large pot over it. I also add about 1/4 cup of dry milk to the mixture.
I freeze some yogurt in an old ice-cube tray for my next starter. Only takes a couple of cubes per quart of milk, and it will keep a long time frozen like this. I really don’t know how long, but I have kept it for months and months and it was still fine.
After I have made it a couple of times from the same starter, I buy a new container of yogurt for fresh starter.
I haven’t tried making yogurt, should one of these days.
We do much the same with the chicken’s buttermilk, start with cultured buttermilk, add some to regular milk and sit it in a warm spot.
Your method works and would be cheap too.
When I was a kid, and we did not have refrigerators our milk would turn to clabber, which we ate as folks do yogurt today.
I do not know if it had a culture in it, but suspect that it did.