Hi & welcome,
I think while genetics do occasionally play a part - there are certain lines of badly bred show horses - most domestic horses have thin soles and weak heels to some degree or other(just saw 3 new horses on the one property today with soles that yielded to my pinky pressure!
) and this is due to environmental factors - diet, management, environment, etc. - It will depend on these factors & your ability to change what is unhelpful, as well as the state of his feet at that age as to how much you're able to help him grow thick, strong hoof capsules. Diet & nutrition are a huge factor.
I do not believe keeping shoes on him will help him grow stronger feet, but it will help him be more comfortable & if padded, protect his feet somewhat from further injury, and allow you to ride him on whatever without further effort. I think it depends what you want/are able to put in, to a large degree, as rehab will take effort & time. I would personally choose to at least try rehabbing him rather than just managing as long as you're able.
If he is so bad that he is even sore when bare on soft ground, in his paddock etc, then his feet are severely compromised & he will need protection in this environment too. Pads &/or boots or hoof casting are some good options there. While it depends on the state of his feet & how long they've been like that, as well as lifestyle & nutrition, as to whether he can become really strong footed/thick soled, it is rare that they can't be made a lot better at least. You may always need to protect his feet on rough surfaces though.
He also went to the vet just before I picked him up and the vet had dug at a few sore spots and found a stone bruise with a blood pocket right near the surface.
While I'm not a vet & obviously haven't seen your horse, and cutting open abscesses is a common practice & sometimes may be appropriate, I'm skeptical about this for a number of reasons. Firstly, opening abscesses is different to digging at 'sore spots' & bruising. If the horse is so thin soled, of course he'll have many 'sore spots' when direct pressure is applied with hoof testers or such. Of course if he's not protected & made to go on hard/rough ground he'll likely have bruises & possibly worse. Of course if he's that thin soled, live tissue/blood will be very close to the surface. Digging into an already very thin sole will obviously remove even more protection. It will open the live hoof to the environment, making serious infection more likely, and using strong antiseptics can ****** the growth of the solar horn. As this was done, he would need bandaging, cleanliness & wound care as well as just padding, until the wounds have healed.
My other 2 horses have such amazing and healthy feet that I only have to have them trimmed once a year!
That's pretty unusual. We're discussing 'self trimming' on another thread here. Most domestics don't have such a productive environment to do away with trimming. Your horses must do a lot of miles on varied, abrasive terrain. Is the once a year after your annual holiday by any chance??