I've been dealing with ECU tendinopathy since 2022 after overtraining (board climbing + hangboard without enough rest days). I got no effect from a steroid shot except a temporary pain flareup. Orthos love to give cortisone shots but it is now becoming understood that "tendinitis" is often a misnomer because minimal inflammation is involved; instead tendons go through a process of reactivity and degeneration when consistently overloaded. Appropriate loading, conversely, strengthens/remodels/rebuilds. Cortisone suppresses inflammation, so it can help with many issues but often doesn't have lasting benefits. Also, research suggests it can interfere with tendon remodeling so it can actually be counterproductive for tendon issues.
What did help me was to stop pushing through the pain thoughtlessly and gradually re-introduce stimulus including climbing. I found a good hand therapist to help me through this process. I did pretty standard rehab, most importantly strengthening wrist extension -- again very gradually. Good physical/occupational therapists always seem to use the motto "let pain be your guide." If it hurts during, after, or the next day, it's probably too much for the moment. Otherwise movement/loading is generally positive. Total deload can lead to atrophy — the tricky part is striking the right balance. (I personally made the mistake of going too far in both directions.)
By the end of 2022 I was climbing again, and have continued to do so without major issues. I don't rehab anymore apart from warming up for climbing with bands. I did find that computer work continued (and continues) to irritate it, though actually climbing sometimes makes it feel better now. I just started a course of ESWT treatments, which have some good research behind them -- the jury's still out on that one, but I can report back.