Is uninstalling an app and reinstalling it over and over again bad for the Xbox

No it won't cause damage. More annoying for you to do than anything else.

As already
explained in your other thread about the TMX app, you don't have to worry about apps damaging your console.

The only way you could harm hardware by repeatedly uninstalling and re-installing an app, is if you use an SSD drive to install and uninstall the app. But in order to actually cause the SSD to fail because of that, you would have to live longer than any
human being and constantly uninstall and reinstall the small app over and over.

You see: A SSD drive has storage cells that can be changed (i.e. "written to") a few thousand times before they eventually break. SSD makers know that, and make the memory controller so that they distribute writes across as many free storage cells as possible.
And you actually get more storage than advertised, so even after some cells break after some years of constant writing, they just activate cells from the spares they have on the drive.

So in order to "break" a SSD you'd have to write several Gigabytes EVERY DAY for more than 10 years on average to wear out an SSD. And with an app so small as the Background Music Player, you can't even achieve the amount of writes to get to a Gigabyte each
day.

So, if it makes you feel better: Install and uninstall the app as often as you like. It won't hurt your console.

But rest assured: As I explained in your previous post: Every app has only VERY LIMITED access to the Xbox System as a whole. There is no need to be afraid of an app. They can't hurt your console or you in any way, as every app and game is running in a simulated
hardware environment under a Hypervisor.

Every app you get from the Xbox Marketplace has to undergo rigorous testing and verification by Microsoft. If they find malicious code or behavior, the app wouldn't be allowed on the Store in the first place.

So every app you get through the Marketplace is as safe as it can be for your console.

If you still have trouble trusting the Background Music Player App, you can even

contact the maker of the app through their website.

there shouldn't be a issue with it going on the console size and game sizes the console was built for u to spend more money on external drives or deal with agro if rotating your games so its very much what the console was made for so damage is very unlikely
unless its 10 years old and taken insane damage from say being stood on by elephants

It is okay to install and uninstall as you wish, this is how people with 500gb consoles rotate their games.

I had to do pretty much the same thing until I purchased an Xbox one x with a 1 terabyte memory.

Never reinstalling game or app make damage or hate the console dont worry about and what about to install soundcloud its better than USB you can play just with internet and when you play a game you can listen too