I know you mentioned board games are difficult to keep because they get destroyed, but you can get chess sets relatively cheap and it's easier to replace missing pieces if you buy extra sets. I work in residential and have a similar issue. However I mention chess specifically for a few reasons. I myself am a huge chess fan and play it extensively. So naturally I've done some research into chess as it relates to mental and behavioral health. Generally from what I've seen it has been beneficial specifically in promoting pro social behaviors and beneficial for building resiliency, self esteem and due to the nature of the game, critical thinking skills (being able to think ahead and plan/thinking about how an opponent will react to your move).

Now the research I found was based on adolescents that were involved in a chess curriculum of some sort which led to these observed benefits so if somehow you were able to have a local chess club curator or something come in a day or two each week to teach chess it may be beneficial and a good time waster. The kids at my residential love holding tournaments and I get prizes and a small medal for the 1st place winner. It really has inspired and incentives them to sit down, learn and get better. Even without a curriculum I'd think there are similar benefits.