Why does infill percentage stop the print from sticking to the bed?

The more infill, the more material. The more material, the more stress is inside the part while it cools down from printing temperature to ambient temperature. Parts with higher infill density tend to warp more (the edges curl up).

But 20 % should be fine, you shouldn't have any issue at that percentage (unless you're printing with ABS/ASA).

I think it's a first layer issue, the 0.2 mm first layer gets more 'squished' onto the bed, thats why you get better adhesion. I'm using PrusaSlicer, every default print profile in PrusaSlicer uses a 0.2 mm first layer, maybe there is something like that in Cura too?

For example the 0.1 mm PrusaSlicer profile will squish a 0.2 mm first layer onto the bed and changes to 0.1 mm layers for the rest of the print.

Basically you have 2 issues, first, an adhesion in combination with layer thickness problem, second, an infill problem.

Starting with the infill issue, when you lower the layer height, without increasing the amount of layers for the "Top/Bottom Thickness", you get a very thin shell (unless the top bottom thickness is expressed in mm). A lower layer height should, because of the lesser amount of filament being extruded over the infill, should be accompanied with a higher infill value, but that is necessary for the top layers, your issue is with the bottom layer and adhesion. As said, a lower layer height also implies lower filament flow, for the first layer this lower flow causes an inconsistent flow to adhere the filament to the bed (probably caused by the gap between the nozzle and bed from leveling with a piece of paper). Most slicers will add some extra features to increase the change to get the filament to stick to the bed; one of those is an increased first layer height (e.g. in Ultimaker Cura, the first default layer height for Ultimaker 3 printers is laarger than the rest of the layers), others include modifying the flow by e.g. over-extruding for the first layer.

You could try to increase your first layer height to the value you create successful prints with, specify the thickness of the bottom and top (or increase the amount of layers for printing these) and increase the infill percentage.