I can't see any reason why not. I think it's a great idea.
You say your looking to build it up, what parts do you have? A lot of seven speed stuff is still out there, on NOS or good condition used. One thing I'd suggest, when using a single ring up front it's good idea to put a chain guide mounted on seat tube, something like this: http://cyclocrossworld.stores.yahoo.net/3rdeyechwa.htm
If the terrain you will be riding in is not too extreme, you should be OK. I have several road bikes that use 7 speed cassette rear hubs and I think they are great, although I use triple front chainrings. Even so, I rarely use the granny gear and run the middle 36 tooth ring about 80 percent of the time. I have 13-32 cogs on the cassette. That gives a pretty good variety of gears. If you put a 38 chain ring on the front you should have enough gears to get you though flats and moderately rolling terrain. Just be sure to line up the front chain ring even with cog #4 on the 7 speed cassette (so your chain line will be OK when using any of the 7 rear cogs). On long downhills you will run out of gears but no worry, just tuck and coast as if you are riding a singlespeed. If you need a new cassette try nashbar, they are selling SRAM clone 13-32 7 speed cassettes under their housebrand for $15 and they are the cat's meow.
I rode a mnt bike with a broken FD for a while. I could only use the middle ring and the granny but I had to stop and move the chain by hand (the third ring was bent). my RD wasnt set right so I only really had 5 gears in the back.
The success of your project depends mostly on your terrain, your love of cycling, your happiness with the fit/personality of that bike, and your ability to operate in a wider range of cadences.