And when the unique cases happened when you are not in home then what you will do ? That is why I'm asking buddy. Everyone has different reasons, right buddy?
First, like others said, signing a message requires a private key of the public address and if you use an online website to do that, you leak your private key. When you leak it, it will no longer be your private key. You can lose your bitcoin by that website owner, very likely to lose it because who will ask you to leak a private key, maybe only scammer.
Next, about signing a message to prove something, mostly it is to prove that you are a real person, not an impersonator, hacker, when you are dealing, trading with another person. Mostly it does not happen and you won't be in that situation.
Now, if you need to sign a message on mobile, you must - in my opinion, have two wallets, two public addresses and stake them somewhere, one on computer and one on your mobile. When you are hanging out, on mobile, you can sign it on mobile with that mobile wallet. I don't recommend to import your wallet on computer to your mobile, just to sign a message. It's risky.