There turned out to be two issues with this choice. The first is that Vega has a very unusual spectrum for a star. This means that more normal stars appear to have weird differences in their magnitudes between different colors. But it's not the stars themselves that are weird, it's just a weird choice of zero points!
A more serious issue became apparent when CCDs became more common in the 1970s and 80s. It turns out that Vega is somewhat variable. You can define the zero point of the magnitude system to be the average brightness over a long period of time, but that doesn't really help you on any given night since the equipment needs to be calibrated daily (or more frequently --- temperature and atmospheric changes require re-calibration).
Another source of annoyance here is that Vega is also very bright. This was a benefit in the days of photographic plates. But modern telescopes with CCDs cannot observe such a bright star. It almost immediately. So this makes calibrating the equipment trickier. (You essentially need a two step process where you use a small, specialized device to calibrate against Vega and then measure the flux from a dimmer reference star, and then measure the reference star with your telescope.)