Tallow is a good soaping fat. You must not have done a lye discount which is why you had great laundry soap.
Yup, that was the first soap I ever made. Since than I made plenty of smaller batches using Crisco, olive oil, coconut etc. I was forced into making my own because all the solids they put in soap was reacting with my water and making an incredibly hard cement on and in my plumbing fixtures.
I buy the liquid soaps now, but have 5#s of home made in storage.