Injured through the remainder of the season

One of the players on my team is pitcher John Means of the Orioles. After having pitched in two games, Means is going on the 60 day DL in order to get Tommy John surgery and will not be pitching again this season.

https://www.mlb.com/news/john-means-to- ... 20campaign.

Means already has a 6 injury rating, so I've already learned to my dismay about the potential injury on a 6-7 roll.

I get the impression that although Means will not be playing again this season, his Daily card will continue to be available. Is that correct? If so, does he basically have the same Daily card through the remainder of the season, and one just takes a chance of frequent injuries if you have him on your team?

Correct. His card will be available all year, and it should stay (more or less) the same as it is now.
Just FYI, in addition to the "frequent injuries" you mentioned, his 6 rating also makes him vulnerable to the dreaded 60-game injury.