relays with built-in LEDs?

It is possible but not really financially viable, some of those relays pull high current adding more bits to it would make the relay larger which manufacturers do not want, plus most cars already have inbuilt warning systems anyway, its just another level of shift to go wrong.

Not all that difficult to butcher a dead relay and put a light between the coil contacts if you really need one (I would be inclined to use a filament bulb because I'm not sure if you can rely on the polarity always being the same, although there is probably a standard).

If you simply wire an LED across the coil, you have no proof the relay has made. Just that there is power to the coil.

To wire an LED across the contacts restricts the uses the relay can be put to, as LEDs are polarity sensitive. So you'd need a different type for one which switches ground, to one which switches +12v. And that's before you consider many relays are changeover types, etc.

To have an output indicating LED would require the LED to be in parallel with the load. That will need an additional terminal for either a ground or earth wire. Your 5p LED is now going to cost £2 x number of relays in car. £50 additional cost price? £100 on the retail price? All for something that 99.999% of owners don't want. The function of the device (fuel pump buzz as it primes, rear screen demists etc.) shows the relay has made.

Unplug your relay, stick a 1.8W dash bulb in the coil terminals. A 21W bulb may blow the low current transistor that switched the relay.

The coil may already have that. Depending on how it is switched. Or not.

But you're correct - it's a silly idea.