It seems that you're using an outdated browser. Some things may not work as they should (or don't work at all).
We suggest you upgrade newer and better browser like: Chrome, Firefox, Internet Explorer or Opera

It is fixed by restarting the client, but it seems really strange that a feature completely disappears after a system sleep.

I've noticed that once I power the system back on, the client briefly says that I'm offline and some features might be unavailable. Well it's just a momentary hiccup with the connection, I'm definitely online, but the stacking still doesn't work. Pressing Sync Integrations doesn't fix the problem, the only option is to restart the client.
There really needs to be a way to manually stack games as it would be a workaround for both this and for when the automated system fails.

I've mostly encountered the latter where the games remain unstacked as the automated part of it is skipping them.
Is that what the issue is? I thought game stacking had be removed, and had come here to check.