What are the advantages of a internal high-precision oscilator over an external one
If I have a lab with multiple devices, (why) would I still want a separate precision timebase (such as a TCXO) in every instrument, instead of just getting a stand-alone reference and provide all of my instruments with this reference?

My answer would be: flexibility

At the moment I work at a small company and we only have a few measurement instruments which would need an accurate clock. We also have no need for a more precise clock so we don't want to spend money to buy one.

Also we do not need very precise frequency synchronization, the few ppms frequency difference we get from each instrument using its own clock is perfectly acceptable to us.

So we do not have and do not need a common reference clock.

Couple of years ago I worked at a larger company where we had lots and lots of measurement equipment and there it was standard practice to use an external clock. Obviously the money was already spent on this 10 MHz source so we better use it as much as possible. This also helped avoiding small frequency differences between setups.

Nice thing is, you can do both with the same set of instruments. And since  for most equipment an internal oscillator does not add that much to the price I think it makes sense to include one by default.