Is there a trick for tightening stubborn fasteners in restricted spaces?

Wobble extensions and a magnetic socket sleeve. That combo solved the exact same problem for me on an old lathe and a bunch of tractor guards where the bolt holes are barely bigger than the fastener itself.

Here's how I do it. Stick the hex bolt head-first into the socket so it's held by the magnet or a bit of that green sleeve material some sockets have. Feed the whole thing thru the tiny access hole, wiggle the wobble extension untill you can feel the threads catch, then spin it with your fingers first. Once it's got a few turns on it, you can put real torque on without the bolt wandering or cross threading. The wobble joint lets you come in at an angle when the hole doesn't line up straight with the bolt, which is most of the battle. A set of three extensions, the 3, 6 and 10 inch ones with the swivel ends, runs about the price of a lunch. They're not as rigid as solid extensions so I wouldn't use them for final tightening on anything critical, but for starting bolts through a pinhole they're unbeatable. Also, if the bolts keep stripping, check you're using six-point sockets and not twelve-point. On older machinery the heads are often allready rounded from decades of previous owners mauling them, and a twelve-point socket just finishes the job of ruining them. If a head is already chewed up, a slight tap with a hammer to seat the socket deeper can buy you enough grip to get it moving. I keep the short wobble extension and a 10mm and 13mm socket in my pocket for these jobs now. Went from twenty minutes of swearing per bolt to about thirty seconds.

Another trick worth trying: thread the bolt into the socket backwards a couple turns so it sticks out of the socket like a little handle. Use that stub to feed it through the hole and get it started, then pull the socket off and switch it around the normal way once the threads have bitten. I've done this on a 1958 drill press where the access holes were maybe 5/8 wide and it saved me alot of fumbling.

Also a cheap ball-end hex driver helps if you can get the bolt started but need a bit of angle for the final spins.

That backwards bolt trick works on a lot of old machinery, can confirm. I did the same thing reassembling a bandsaw wheel cover last winter, access holes were tiny and the bolts wanted to drop constantly. Only thing I'd add is to snug it up just finger tight before pulling the socket off the second time, learned that after dropping one inside the cabinet and spending twenty minutes fishing it out with a magnet on a stick. Which I now also keep in the pocket.

So the bendy stick thing basically lets the socket go in crooked and still spin it? I tried using one of those long screwdriver-looking things I found in my dad's old toolbox but it just slipped off the bolt every time, so maybe that wasn't the right shape. Does the bendy part wear out or get loose if you use it a lot? I don't want to buy a set and have them go floppy after a month.

To answer your question, the swivel end does loosen up a bit over time but mine have held fine for a few years of weekend use. They get sloppy faster if you use them for final tightening on big bolts, so I just don't. Light duty starting bolts, they'll outlast the truck you drive them around in.

Also if the one from your dad's toolbox slipped off, it might've been a speed handle or one of those flat flex shafts meant for screwdriver bits, which is a totally different animal. Try a proper socket on one before you write the idea off.