What is a tried and true method of propagating a rose from a bud or stem from an existing rose bush.?

Not commonly done these days by nurserymen, but you can often get cuttings to root - especially from ramblers. They graft a lot of the bush/hybrid tea types because the natural roots are easily affected by diseases, but you can get away with a lot if you are not surrounded by black spot.

Take a cutting of fresh growth with about an inch of woody growth as spring turns to summer, dip the butt in rooting powder, and shove it an inch deep into a 5" pot of soil. If you work like my Dad, split the butt end by cutting about 1/2" up with a sharp knife. Water occasionally, but otherwise ignore it until next year. If it shows buds next spring, wait until you have young leaves and plant it out.

I've used this guy's method and have had good results:

https://aggie-horticulture.tamu.edu/southerngarden...