Odds of Getting In
If you want to become a dog trainer, become one. But know that it's a hugely popular career when it comes to working with animals. Consider branching out a bit and, again, acquiring more training and qualifications will open more doors to you.
Another thing to consider is that about sixty-one percent of animal trainers today have standard full-time jobs, and earnings are low (source). Turnover tends to be moderate but, as you've already figured out, this industry varies quite a bit—from store employees to Hollywood's go-to trainers.
So you know, do what you love (but only for as long as you love it and it allows you to pay the grocery bills).