Just discovered that I have two options.....Somewhat embarrassing after so many years!
Are you alternating with both according your goals at the time of the shoot or, set on one mode all the time, regardless ?

My camera is manual shutter only, older tech. Canon T2i, I think they are at least on T6i now?

Can't keep up, no reason for it.

Seems like electronic shutter would be quieter and sync faster, both useful.
I'm certain it eats more battery too, the tradeoff.

Depends.

On Olympus with flash I cannot use electronic shutter as it will not fire.  It only fires flash on mechanical shutter.

If I go to electronic shutter, the speeds go up to 1/32,000 second, much faster than the top 1/8000 sec. with mechanical shutter and also using that for HSS flash.

I've used electronic 1/32,000 to freeze hummingbird's wings in bright sun.  Below is a series I ran from 1/250 up to 1/32,000 second.

If I m doing an outdoor timelapse, I will use e shutter to save on the mechanical shutter.  There are some limitations with e shutter such as banding under some artificial lights.