The opening through the cuddy deck should be shaped so that you can drop the mast at the correct angle (tilted just slightly off the vertical).
I inherited the setup that I document, but I had to rebuild it due to the original wooden block failing over time (rot or splitting or both). When I did that, I ended up with the entire sleeve so tight that the mast would not slide through it, so I had to re-file the groove. A slightly wider groove is also more forgiving when the hinge and the cuddy opening (partners) are not lined up perfectly (what you call "wiggle room").
Fore-and-aft angles aren't an issue, because the hinge accommodates those, but if the groove is way too shallow it would position the mast in front of the opening, not over it - that may have been an issue as well. I have a 4ft or so mast section that I use as a mast support when trailering. That's been a great tool in checking tolerances etc, because other than a shorter length (and therefore easier to handle) it's just like the lower end of the mast.
I don't stand on the cuddy

If I did, I would use a spotter.
As it is, I use a spotter when I have one, because I usually lean over too far to see the mast step. However, the partners will hold the mast well enough that I can let go one hand and take a peek - especially if sound tells me I'm nearly on the step and just need a bit of a wiggle to make the mast foot slide onto the mast jack.
Having everything aligned so that the mast can only go down in nearly the correct angle helps as well.
With time, I got better at aiming....