Don't (never) import wallet mnemonic seed from your current wallet like Metamask wallet to a new web wallet like Bitget, Phantom, OKX. Some wallets have that import feature for you BUT NEVER choose Import.
If a new web wallet is bad, has back door, you can lose your money.
I think this is a bit paranoid imo.
It is not, in fact, there are lots of cases where a person losses his funds on his web wallet because he carelessly inputs his mnemonic seeds or private key to import our wallet. Even linking our web-wallet that authorize the site to access, modify or do an activity to our web-wallet is dangerous since we never know if they have that ill-intention towards our funds.
Every web wallet is basically the same, and those popular wallets do not have backdoors, this is very unlikely.
Indeed reputable wallets tend to have no backdoors but in this case @OP is talking about new sites that requires a person to link his web wallet. But I would say not all web-wallet are the same because there are some wallet that were released with malicious intent.
Anyway nobody should hold more than a few dollars in any web wallet.
That should be the case but many of us are too lazy to move huge amounts of fund to offline storage especially when a person is active in trading on swap platforms. So it does not hurt to be reminded to take more precautionary steps in dealing with new websites that requires linking our webwallet.
I have a wallet which I use the same seed in metamask and another wallet in different devices. This is not a big deal imo.
It won't be a big deal until one finds his wallet emptied by a hacker. It does not hurt to be more careful when dealing with new websites that require us to link our web wallet to their site, IMO.