Exact process will depend on the engine design. There are strobe-light setups that can be used to check. If you can get at the belt to do a visual inspection, that can help. Or if it gets a certain kind of noisy, but as an amateur, I've always had trouble picking out the bad noises from the good noises.
A 300k timing belt is, in all likelihood, a timing chain, not a belt. Although I suppose you could get that kind of mileage out of a belt if you drive really, really gently and put on 75k miles a year or so. But even sitting still, the rubber eventually rots.
Well, first it was $775, then $2k, now $3k. Yes, a $3k repair on a $2k car would be silly, but that's engine or transmission replacement territory, which is usually when a car is considered totaled anyway. A $500 repair on a $5k car generally isn't contraindicated, unless you intend to sell it fairly soon. If so, you're right that you won't get that money back. But you won't get to keep it all, either - unless you find an absolute sucker for a buyer, a car that wasn't taken care of isn't worth nearly as much on the used market as a car that has been well-maintained. Any decent mechanic doing a pre-sale inspection will spot (most of) the deferred repairs, and the buyer will adjust their offer accordingly.
If you don't intend to get rid of the car, the best value from a total-cost-per-mile-driven standpoint is usually obtained when you drive a car "into the ground" - 200k or more. 300k now, maybe. Or 500k. But you have to sink some money into maintenance to get there. Yes, maintenance costs do go up as the car ages and you hit certain intervals (particularly the 100k or 120k services) but once they're done, you're good for another 100 or 120k miles, and it's more than outweighed by not having a car payment, lower registration and insurance premiums, etc.
Also, humans are irrational. Apart from the obvious sunk cost traps that occur with cars, there's a "devil you know" component here. A person generally places a value on their own vehicle that's significantly higher than KBB, because they know what's wrong with it and know what's likely to fail next, if anything, which gives them a sense of security. When your $2k beater drops its tranny into a pothole, if you buy some other $2k beater, you're gambling that it won't do the same thing next month. Which is why when they DO get the $3k repair bill for the $2k car, most people who aren't completely strapped for cash will buy a newer, more expensive car - because they perceive it as only equally reliable to their old car, and not another $2k beater that's an unknown quantity. So realistically it's a choice between a $3k repair for a $2k car vs. a $6-10k car.†
†Unless, like I said, you're just completely broke. In which case, good luck and god bless. But now you're dealing with a situation where a new $500 car every 6-12 months and NO maintenance is often the cheapest option that works within cash flow. And yeah, people do that too.
And if we're talking about a $2k car that has some sentimental value, some people actually will spend the $3k on the repair. e.g.:
IMO, though, regular maintenance is basically a form of insurance. If you spend the money now, you're less likely to have a catastrophic failure later.
Cars will put up with a lot of abuse, up to a point: you can drive on tires until the cord is showing, go 12k miles between oil changes, let the tabs of your brake pads carve a furrow into your rotors, never change fluids, never replace something that isn't completely broken, and otherwise save a lot of money in the short term by not putting a single penny into your car that isn't absolutely necessary to keep it running and driving. But then you're more likely - not guaranteed, but more likely - to have a major problem at the least opportune moment, and can get hurt in the process. Gamble if you want, but try not to be driving anywhere near me when your wheel falls off because you didn't get that "wub-wub-wub" sound fixed.
Deferring maintenance in this fashion may also result in more costly repairs when you do finally fix things, as regular replacement of some stuff (e.g. engine oil) will usually help other more expensive parts (e.g. engines) last longer.
tl;dr - take care of your shiznit and it will take care of you.