Do you think therapists get tired of dealing with people who are just unhappy with their job in a bubble?
Do I think that your therapist would judge you for your problems? Absolutely not, if they are even an OK therapist. They are professionals, and are not trying to be your friend.
Another thing. The first therapist I went to sucked. She had me doing worksheets and other stuff that just didn’t work for me. I ended up switching (my wife made me switch) to a new therapist and it was a huge difference. If you do end up trying therapy, remember that this is a professional situation. You wouldn’t go to the same bad sandwich shop for lunch every day. If your therapist isn’t working for you, and you have given them a honest try, just switch therapists.
Ultimately, therapy was a leap of faith that I’m glad I took.
Being unhappy for any reason matters. I discovered I was unhappy for much more valid reasons than I ever imagined and taking that leap to therapy was quite life changing. The key thing was that my burnout had legitimate causes that weren't really my fault or a weakness or flaw in myself (at least not that I could easily control), and I learned a huge amount about how I needed to move forward.
I really don't think I would have done it, ever, without therapy.
I had one in my city before and he was great too, but he knew a colleague might have more relevant experience. I was doubtful about the distance aspect, but it's been totally fine since day one. It's definitely worth considering.