Getting blog pagination to work on page set as front page

Yes, you’re correct that on the static front page, we use get_query_var('page') to get the current page number.

And actually, the issue in question (“the number 2 isn’t highlighted“) happened because you used get_query_var('paged') (note the paged) with the current arg for paginate_links(), which means you’re passing the wrong number.

'current' => max(
    1,
    get_query_var('paged')
),

So all you need to do is replace that with 'current' => $paged, and the problem will be gone.