My suspicions are if you are having significant difficulty, the screwdriver you are using does not have the proper point to it. Tiny phillipshead screwdriver isn't really a good description of what you need lol.
There are actually numerous different "tiny" sizes of phillipshead. A good set of precision screwdrivers (such as you might use for phone, or tablet repair) should have a #1, a #0, and a #00 phillips driver. You can likely get away with the #0 if that is what you have, and you work very carefully, but the correct size for these little buggers is the #00. If you have a #1, you likely won't be able to do anything but tear them apart.