Very short-pull derailleurs designed specifically for 1x6 drivetrains? EDIT: After some searching, it appears this type of derailleur is currently available in the form of folding bike derailleurs, such as the Shimano RD-FT35. It is a SIS 6/7 speed compact derailleur with chain capacity of 17 teeth, max cog size of 28t. As a bonus, it's very cheap. It seems to be targeted as an OEM item, and details are a bit hard to find, but appears to be perfect for this application. I ordered one to try.

Was there ever such a thing as a rear derailleur for a single chainring, but a 6 or 7 speed freewheel? 

What I want is a derailleur optimized specifically for 14-28 or 14-34 freewheel, but a single chainring, and indexing compatibility doesn't matter.

The reason I ask is I fix up lots of cheap bikes that are being thrown out and make them rideable again. It's become almost a vocation at this point. In the case I get a good frame and wheels, I even try to make them nice.

I usually throw away the front derailleur and go with a single chainring, for cost and simplicity. That is, when I don't outright convert to single speed, I only keep the rear gears. I ALWAYS change to friction shifting, so compatibility is not an issue.

The common freewheels still available seem to be Shimano 6 speed 14-28 or 7 speed 14-34 "mega range" freewheels, or generic equivalent freewheels. 

Since I don't need the range of chain length from multiple chainrings, this means even the 6-speed derailleurs, which are designed for 3x6/18spd gearing, are overkill in terms of chain pull, and result in a chain longer than it needs to be. At the same time, I need a derailleur that can shift onto the 28 or 35 tooth sprocket of the freewheel. 

I would really like a derailleur with a shorter cage just for 3-speed to 7-speed freewheels only, with a single chainring. Are there any rear derailleurs, new or vintage, that are like this? If not, what would be the closest?
In theory you could get a short-cage derailleur, but that would add significantly to the parts cost and not really be much better.

Back when I did a lot of fixups, it was all about making the bike as good as it could be, without spending anything on the more expensive items.  We'd often replace brake pads and cables with new, but items like cassettes and derailleurs were very rarely bought new.

Fixups are a different mindset to working on your own bike.

Instead, we'd knock down bikes that were too-far-gone, and put the usable bits into storage bins.  During downtime, volunteers would clean and recondition components, then return them to storage.  So later, if someone's fixing a bike and finds they need a rear mech, they'd pull one from the parts storage bin for rear mechs and fit it up.  The old part would be either cleaned and serviced, or scrapped for metals which earned a couple hundred dollars perhaps twice a year, but each time it was a truckload of pre-sorted metals.

The expertise comes from deciding how dead a bike has to be to go to parting-out vs what's worth saving.  That is affected by how busy the operation is, what parts are low-stock.  Your local area also affects the supply line - there are always old bikes turning up.  And finally the ultimate limiter is how much space you have for storage.