Definitely concur with the trying him at center-back bit. I look at Callum Chambers who also played at right for Southampton at a good level but some of his best performances have come at the center of Arsenal's defense, and Shaw's overall skillset is pretty comparable with him. That's the beauty of having young players, their positions are very fluid at times and they can unlearn+learn newly acquired concepts at a faster rate.
All things point to him being very good at the defensive aspect already contrary to media opinion and this calculated calmness is something we lack between our current set of CBs. Give him a run of 5-10 games, if it doesn't work out then let him carry on at LB. And like you pointed out, there's an abundance of quality young LBs around, so replacing him won't be an issue. We could get someone like Kurzawa or Rodriguez for a relatively smaller fee than an equivalent top class CB.