Let’s see, lots of money and plenty of time off.
Many STEM professions would fit. They tend to pay well and have regular hours, but not necessarilly a lot of time off.
Teaching is great because, as @Foxglove mentioned, it allows the time. Don’t forget other jobs (administrative, clerical) working for colleges or school districts which also allow summers off. The pay can be too low for comfort to support an expensive amateur career but with careful budgeting can work.
Some medical specialties where work can be scheduled and emergency calls are very rare or nonexistent would also fit. I’m thinking of the doctor who just did my colonoscopy. He schedules all of his surgeries for two days a week and has some office hours, but I think he is off Friday, Saturday and Sunday. Orthodontists and ophthalmologists tend to have pretty routine schedules. I’m sure there are many other medical professions which fit the bill. The con to these professions is that they require a significant number of years dedicated to education during which time it would probably not be possible to pursue a serious equine career; it would have to be delayed.
A friend works for a government agency which has a four-day work week. The pros are that there is a pension and plenty of time off, as well as a four-day work week, but the con is that the pay is less than in private industry.
In the case of having a medium-paying job and plenty of time off, it’s probably a given that many women who are heavily involved in horses have a working spouse or partner who shares some of the living expenses which frees up more discretional income. I hate to say it, but the reality is that marrying money is also a way to have support.
Two factors are really important for financial security and success, and I think they are worth mentioning to any young person asking about their futures, because really, if you budget well you can accomplish your financial goals with most jobs. I like to mention both things to young people and emphasize the first to young women since it affects them more. That first one is unplanned pregnancy/giving birth at a young age before they have their education and some financial stability. This will almost surely delay their financial success and progress. The second is addiction. I tell them to look around at adults in their lives who are successful and compare them to those who aren’t. My observation is that those who are basically failing at life are often addicted to something harmful: alcohol, drugs, compulsive shopping, gambling, etc.