I'm seeing this sort of interface where grout has been placed between the wall tiles and the door jamb.
Is that the best material to use? Why not silicone? And how about aluminium angle trim? Or is that just over the top and too visually intrusive?
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... is that just over the top and too visually intrusive?
Yes, but that would be opinion; beauty is in the eye of the beerholder
Why not silicone?
Caulking would be better here (not necessarily silicone) because of vibration induced cracking of hard grout (as seen in your picture). The problem with caulk is color matching, that's why I would use paintable caulk and carry the door jamb paint over the caulk and up to the tile line.