Solution for covering carpet glued to concrete on a balcony
It's a pain in the neck but you need to remove and dispose of the carpet. Anything you put on top of it will only cover a nasty mess that will get worse over time.

Once the carpet is removed you will need to clean the cement of carpet glue. There are many videos and guides for how to remove carpet glue.

Once the cement is prepped you have options. The vinyl tile you mention needs to be rated for outdoor use. Sunshine is hard on plastics. You don't want to install something that will be as big a mess as the carpet later on.

Options:
Not a complete list...


Leave the cement plain. This is the lowest effort/cost solution.
Stain the cement. There is a trend toward stained cement. You'll need something that is outdoor appropriate. Lots of videos and guides on how to stain cement.
Outdoor tile - this isn't cheap but it is durable, easy to clean and looks nice.


I like to get an estimate for what it will take to hire work done. This tells me how much money I'm saving if I do it myself - makes me feel good. On the other hand, if the estimate is reasonable I might let them do the work!