Is it safe to connect your wallet?

Connecting your wallet doesn't really do much: it doesn't send any transactions or get sensitive information.

About all it does it gives your wallet's address to the website and the possibility for the website to request actions from the wallet - actions which you, as a user, need to accept manually.

In theory, everything would be smoother and more user friendly if your wallet automatically connected to a site whenever needed.

But the connection is not automatic for security reasons: by auto-connecting (or even just by telling the site that you have a wallet) you would give too much information to the website. A malicious website might try to detect whether a user has a wallet, and if does, then the site could start various crypto scams / phishing attacks. If the website has no idea whether you even have a wallet (as is the case) unless you connect your wallet, the website has no idea whether it's worth it to even try some sort of nasty stuff.

In general connecting your wallet to any website isn't a big security risk, but it's simply better to not connect unless you have a need for it.