Definitely go with Anker this is the one I’d recommend. Plugs straight into a wall to charge and can act like a charger.

There are 2 other issues. This occurs on some, but not all USB battery packs. I will quote straight from Adafruit.com.

"When you start and stop charging the pack, it will flicker the output, this can cause a 'power sensitive' device like the Pi or an iPhone to reset on the power supply. "

"If using it with a low current load, 100mah or less, the pack may fall asleep."

I can confirm a few battery packs from 2 reputable brands that do not have the first problem.

Anker Powercore Fusion 5000/10000 [note the "Fusion" in the brand name]

RavPower 2-in-1 Portable Charger 6700/10000

The second issue will likely only come up if you are using an older Pi or Zero and sitting at an idle state.

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The biggest issue I see with the unit that OP linked is that it does have the pass thru function [meaning charge and power at same time] but you have to hit the button twice to initiate this. This could hinder functionality while using a computer attached to it, possibly causing unnecessary reboots as you put it into this mode.

I leave it plugged in 24x7, yes. I have it plugged into an Anker battery so if the power goes out, I get notified and I can then have a reasonable amount of backup power for the robot. https://smile.amazon.com/Anker-PowerCore-Portable-Charger-Foldable/dp/B01K702S66

I do not let it discharge - it probably would help, but I'm too lazy. The battery life is definitely "not great" - but good enough for my uses. It has a soldered 18650 cell, I believe, so if you were particularly interested, I think you could replace it.

I rarely take screenshots but I didn't have issue finding them on an Android device. I have not tried from my iPhone.