Although I agree it's a good thing to do some research and be critical to what you are told by professionals, you also have to be very critical about what you find on the internet. I've just finished my second year of studying physiotherapy and have seen A LOT of nonsense and very bad advise given on forums like this. I think it's great that people want to share stories, but a lot have changed in 20 years. The stories are still good
but don't expect to go through the same treatment as back then.
I'm not trying to talk bad about anyone in this tread, just giving my own personal/professional advice.
On a more serious note - healing time is different for different parts of your body. Muscles heal the fastest (weeks), then bone (+1month), then tendon and ligaments (+3 months - sometimes +12 months).
As you might have had this fracture for a while, it might also take more time to heal. Even if your L5 vertebra is all healed up after a few months, the ligaments keeping it in place might not be ready to do their job yet. They will have been stressed and made longer while you have had the fracture (because the tendons alone can't keep the spine in place), so you need to give them time to heal up properly.
If you could choose between getting back on the bike sooner, but always have pain riding or wait a few more months then you feel like, but be pain free when you get back on the bike - it might be a good idea to take it slow for now.
Here's a little something to read if you want - I don't know if it describes the same kind of fracture as you have, but a lot of it might be the same.
It sounds like you are dedicated to your PT plan and usually leads to a quicker and better recovery. Keep in there and stay focused.
Good luck and wishes of speedy recovery
Say hi to your PT from Denmark