I'm going to go a little against the tide here and say that you should move on. I don't agree with your coop. Yes, you could put a long seatpost and a technomic stem on it, but it will never feel quite balanced. IMO level top tube bikes should be fitted according to the standards of the time. Also, it looks like the stem is already pretty long, like a 12 or something. 13cm is the usual limit there. What length is that top tube? Looks short. You could use a setback seatpost but then you start messing with the pedaling efficiency/ergonomics. BTW I'm also 6'2" and normally ride a 25". I can and did ride a 24" for an aggressive race fit (too old for that now), but 23" would just be wrong.

For a commuter, I think it would be OK to refit this for a larger person, but for a real roadie bike I think it's sub optimal.

If you do end up putting a longer stem and seatpost, do make sure to get proper extension into the tubes. Raising these above max extension is a good way to ruin a frame.