That Longines. Fantastic.These requirements are made-to-order for a vintage watch, especially given that it is a dress watch presumably not for daily use.
I paid $175 for this 1958-vintage Bulova His Excellency (36mm, 7mm thick, handwind, superb Bulova 10BP movement):
I paid $150 for this mid-60's Longines Flagship, handwind caliber 284 (one of many quality Longines movements from that era), 34mm, 8mm thick:
I paid $75 for this Favre-Leuba with a unique manufacture twin-barrel caliber 253, 7mm thick, 34mm (sorry for the fuzzy photo). This watch was made right before the Favre-Leuba owners (Henry and Barbara Favre) bought controlling interest in SAPHIR, the family holding company of Jaeger-LeCoultre, in case you are not familiar with this now-defunct brand:
I only paid $40 for this 35mm Dong Feng with its excellent pre-Seagull ST-5 movement. It's the only watch in my list that would not have been considered a fine Swiss watch back in the day, but it is still quite a nice-looking watch with a quality movement that fulfills all your requirements:
And I paid $125 for this high-beat (5-Hz) Zodiac SST, which is just over the line of your requirements: Automatic, 9mm thick, 34mm:
All of these were cheap enough to buy a decent strap and have them serviced, and still be within your budget. All of them are better than any new watch you'd get at the same price.
Rick "who has a few vintages that meet all your requirements except price, too" Denney