I don't have an answer for you, but your post got me curious about how my Strawberry was handling this. I rarely use the sort columns feature, so I hadn't noticed it.
But, yes, my Strawberry, (1.2.23) has the same behavior as yours, in the playlist columns. I am using Linux Mint 22.3 Cinnamon.
I also run an older version of Linux Mint 21.3 on another computer, and use an older version of Strawberry (1.0.23) from the package manager. There, the behavior in the playlist columns is what you would expect, ignoring articles like "the" in artist names, and sorting The Beatles as under "B", not "T".
That older version of Strawberry doesn't even have the "Skip leading articles..." options, but everything works fine.
To be fair, the options you referenced are under "Collection" in settings, but one would expect the same behavior in playlists.