Perhaps you interpreted the question slightly differently, I took it as the ball was being thrown upwards but we were only measuring the position (with some device) every 0.1 s. Therefore, you can still assume the ball to reach a max height at some fraction not at 0.1 s, we just don't have a precise enough measurement to catch this so we're forced to round to 0.1 s because that's the precision of the device.
This is a common limitation for modeling, so I'm confused as to why this isn't a reasonable approach in your view.

But aren't we essentially looking for the time in 10ths of a second that is closest to the apex of the curve of height against time. Since the curve is parabolic if we take the time to reach the highest point when v has deaccelerated to 0 then the nearest 10th of a second either side of it is our recorded answer?
I've not touched calculus in years so my reasoning could be a little off.

Yep, good first point. This solution doesn't answer the question at all.
For the second part, weird to call basic algebra "complex physics" before solving the same problem with calculus.

The time is always the same anyway.

Yes this kata answer is off due to rounding approach.