Can a weak battery in your truck prevent your infotainment system from coming on?

I have a 20 Denali I believe the battery might be weak or old even though I had it tested a week ago and they said it was okay, but said it still could be a old battery and it's just testing as okay.

On four different occasions the truck acted as if it didn't want to start as if the battery was weak. I did have to jump the truck off a month ago.

Here's my question can a weak or old battery prevent your infotainment system and your air from blowing out cold air properly in modern vehicle / trucks?