Not able to tighten brake any further, but lots of movement without ‘too much’ lateral pressure. Normal?

The brake is tight enough, maybe too tight.

What I see is more or less normal. The centering action is when the springs push equally on the arms. Pump the brakes and see where it opens to.

If it’s off center after pumping, what needs to move is the piece next to the bike that holds the middle of the spring. Sometimes there are flats that you can use to move it. Sometimes you just have to be careful how you tighen the mounting bolt in back.