Just listened to the recording gain.
I think we are hearing a constant noise, it is unrelated to the volume of the guitar.
Therefore the amplifier is not amplifing any incoming noise problem, the noise is somewhere between, and including, the amplifier IC itself and the speaker.
Maybe it is some kind of mechanical cabinet rattle, this may be a constant volume, but you say you tried it with another speaker and it is the same.
It is possible but rather unlikely the amp IC produces noise when supplying power to the speaker but not the reduced power to the headphones.
You could also try shorting the external speaker jack socket switch, this I suspect does exactly the same thing the headphone jack does, disabling the internal speaker in preference to the external one.
Maybe try again with an external speaker via this socket, it still sounds like a speaker problem to me.