Is there a point at which aluminum drop bars and stem are simple "too old" and need to be replaced?
Yes.
Aluminum has a limited fatigue life. The lower the tensile strength of the aluminum and the thinner the tube walls are, the shorter the fatigue life is.
However, it's not years ridden that matter. It's not even kilometers ridden. It's kilometers ridden hard. So if the handlebar has been used for leisurely countryside cruising on good-quality roads with wide low-pressure tires, it has never seen even one kilometer ridden hard. But if you ride it in roads full of potholes, or in city with lots of curbs, or with narrow high-pressure tires, you can easily accumulate kilometers ridden hard.
Unfortunately, it's impossible to say when a handlebar fails. The reason being that it could very well be that 220 gram handlebar lasts for 10000 km in some usage, and 300 gram handlebar lasts essentially forever (which isn't really forever since aluminum has a finite fatigue life, so it would be something like several million kilometers). How does 280 gram handlebar last then? Very hard to say, something between 10000 km and forever.
What I would do is to:
- Throw away the old handlebar, since the history is unknown
- Buy the best new handlebar you can find. Features to look for: butted tubing, 31.8mm stem clamp area (since that's the area where handlebars are most likely to fail), polished smooth aluminum surface at least near the stem clamp area as opposed to black anodized since anodizing reduces fatigue life, high weight, good-quality aluminum alloy, not any kind of stupid holes for Di2. For me, this is old Ritchey Classic that is unavailable now since they ruined it by Di2 holes.
Note that any kind of inspection doesn't tell if it's likely to fail. When a crack appears, it grows very fast. So to prevent fatigue failure by inspection, you would need to inspect it more often than every 1000 km. This would mean stripping the bar tape and removing it from the stem clamp area, although it's possible to "approximate" by looking only at the stem clamp area which is the area most likely to fail at least on the old smaller-than-31.8mm clamp areas.
While at it, buy yourself a 31.8mm gapless stem and ensure you install it correctly by eliminating the gap at the top so all of the gap is at the bottom. Gapless stems reduce fatigue failures of handlebars at the most vulnerable point. Fatigue happens as a result of tensile stress, not compressive stress, and the most vulnerable top clamp point is where the stress is tensile. A gapless stem eliminates stress concentrations there, making fatigue failure less likely.
As for stems, they are unlikely to fail due to fatigue since stems generally have more wall thickness than handlebars. However, stem technology has evolved and 31.8mm gapless stem has features you won't find on 15-year old stem, so replacement is advisable due to improved technology, not due to age. Also of course you will find stupidly-lightweight stems, since if some component is durable, it's obviously an opportunity for reducing weight, and for such ridiculous stems, what I said doesn't apply, they can and do fail.