Back at the dawn of time I was at the very same crossroads as you are now but chose a different path. Hindsight being what it is, I would like to offer you my opinion.
Rose breeding is a small field that saturates quickly.
There have always been very few dedicated rose breeders who receive private institutional support in the manner of Boeriner, Warriner, Zary, Carruth or Lim. While it is not impossible to work into that type of situation, it is very difficult.
University breeders are fewer still and universities have been cutting support for hybridizers. While they have the advantage of access to cutting edge research and technology, they usually have to keep other irons in the fire because of the need to conduct publishable research.
Government rose breeders (USDA, Ag. Canada) no longer even exist.
Independent rose breeders almost always run their own growing and retail nursery operation, both to support their work and to get their hybrids into the marketplace. Harm Saville and Ralph Moore fall into this category. Weeks, Meilland, Kordes and other big (mostly European) rose houses got their start in this way.
A fairly unique model, and the one I would want to emulate if I were starting over, is that of Zaiger Genetics. Floyd Zaiger was an independent fruit breeder who gained prominence for his unique interspecies hybrids. The company is run by his family now. His philosophy was to use bridge species to migrate desirable traits into commercial hybrids, though today it is hard to tell what they are up to as the Zaiger family is famously private.
No matter which way you go, molecular biology is a key component of your future toolkit. UC Davis is a good place for acquiring these skills. It has the advantage of a close association with the rose industry though this has taken a back seat to viticulture in recent times.
A better choice might be Washington University in Saint Louis. That is where the first plant transformation vectors were developed, and it has close affiliations with many of the top global agricultural companies.
It is possible to go through an entire doctoral progam without ever getting your hands into dirt, so be careful to choose a program that will encourage you to maintain your own hybridizing program. To that end, Texas A&M might be worth looking into. They are relatively small so you would have to work at getting exposure to the very latest biotechnology (by, say, doing a rotation outside the institution at a school like Wageningen), but David Byrne has his heart and mind in roses so you would get a lot of first hand experience in hybridizing. Ralph Moore’s breeding stock is reposited there, too.
Standard breeding lines of roses have pretty much exhausted the potential of their genes. There is a lot of potential locked up in species roses and their close relatives, though, and entirely new avenues opening up through technological means. A successful rose breeder will, in the future, need the ability to exploit both biotechnology and classical hybridizing. However you go about getting these skills, I encourage you to chart your own course.
I wish you success, Shane.