Tranq darts don't work like they do in the movies. The target doesn't drop instantly. Even nature documentaries tend to make them seem quicker than they actually are due to editing for time.

Here's a Youtube video involving tranquilizing an African Wild Dog in real time:

https://youtu.be/k4D2pDk4Yvw?t=35

The shot is taken at the 35s mark. The wild dog runs away and even ends up coming back to continue eating it's prey. By about the four minute mark it's starts stumbling around drunkenly, and then only goes down after about five minutes.

How much damage could a human with a gun cause in that amount of time? Plus unlike animals, humans have hands and are likely to rip out a dart before it's had a chance to inject it's full payload.

Ultimately they're just darts of medical sedatives. A bigger dose could theoretically produce a faster effect, but an overdose can easily kill someone. Unlike in surgery you're not going to be able to have an anesthesiologist to carefully administer the proper dose in real-time and monitor the subjects vitals.

It's already a controversial practice for police to have agitated and uncooperative individuals sedated with ketamine by responding EMTs, and even that can lead to severe complications and death despite the administration being done by medical professionals. Giving police pre-loaded darts to shoot at people without medical supervision would be disastrous.