Three comments/ideas based on having owned one of these bikes and built/fixed dozens: Borrow some 700c wheels and see if the brakes work--just an adjustment check. Your 620 was made when 27" wheels were going out; my. own fit 700c's which could have been by design but also could have been just manufacturing tolerances giving me just the right brake post placement. You will find buying 700c tires much easier and, as good as that range of Treks is to ride, you'll wear out some tires!
Many fine rim brake wheel sets are selling cheap these days, just check craigslist. One of my regular customers has bought a few, the quote from him "They're so twentieth-century that nobody wants 'em." He scored a nice Ultegra6500/Open Pro set for $100 earlier this year.
Helicomatics--fast wearing cogs with tooth profiles that guarantee some bad shifts, parts only available at collectors' prices--what's to like?
Wheels that can use Shim/SRAM/Sunrace cassettes will make your life much easier.