I'd ask a farrier/trimmer and if they say no, go back to the vet and ask for more specific reasoning. Your horse's hooves are decent, IMO. Yes the heels seem underslung, but it takes time to bring them back. (I'm working on that right now with my mare and after 6 months I finally can see progress... I think...)
Some horses do need shoes I think, but if you are hesitant, ask for more opinions and get solid reasoning from whomever you ask so you understand why they say what they say.
And, btw, 24/7 boots as mentioned in a post above, is not a good idea. Very rarely are boots recommended for full-time use.