I am wondering if for a once-in-a-month Bitcoin user it would make sense to install and use Elecrum on a Linux Live system….
(to me this seems more secure than to use a Bitcoin wallet-software on my rooted (but well maintained :-) mobile)
You'll have to decide this for yourself.
It definitely would protect against some attack vectors.
Whether the decrease of usability is fine for you, is up to you.
Does it make a difference in such a scenario, if my private keys are stored in the wallet-app or in hardware wallet?
Yes, definitely.
From a usability point of view, a hardware wallet is preferrable.
And from a security pov it really depends on how you manage your linux system (i.e. is it completely offline and air-gapped?)
A completely air-gapped cold wallet system is considered to be more secure than a hardware wallet due to the decreased attack surface.
However a simple live-booted linux distro is not as secure as a hardware wallet.