Same is happening to me. Recently updated firmware with Xbox Accessories app. Unsure if it’s related.

Yea same

Can you try replacing your disposable batteries or charge them if you're using a battery pack?

Originally posted by austinp_valve:

Can you try replacing your disposable batteries or charge them if you're using a battery pack?

Swapping in freshly charged battery packs, or new disposable batteries does not fix the issue.
The Steam Client needs to be completely closed before the controller will remain powered on once this happens.
Correction: see next response

The same old (or new) batteries can be used as expected once the client is restarted.

I just noticed this happens not only when Store is selected.
The wireless Xbox controller will immediately turn off whenever any selection is made that constructs a new window with a visible URL at the top in place of the normal search bar.
The Store selection in the side nav, any Game Info links to an individual game's Store Page, DLC, etc. all cause this behavior.

As long as this window is alive, the controller will turn off immediately when trying to power it back on.
Navigating away from this window (with a visible URL in the top bar) will finally allow the controller to reconnect and stay powered.

Edit: I'd also like to add there is a very brief window when the controller is actually connected and seen by Windows before turning off.
I confirmed this in both Win10 Control Panel Devices and through MS' Xbox Accessories App.

However, trying to reproduce the above behavior is not consistent, as it seems to be completely dependent on an unintentional controller shutdown happening first. On a recent client restart, I could not reproduce any of this until it happened to me again some time later.