How can I keep a screw from slipping off a screw bit?
As for other options: I really like using square drive bits and fasteners (a.k.a "Robertson"):



They have several advantages over Philips-style screws:


The screw easily attaches to the bit and generally stays put (this is the attribute you seem to be looking for)
The square drive bit does not "cam out" if you don't apply enough pressure.
The bit and fasteners are much more durable, which means you don't generally need to worry about stripping the screw head or damaging the bit.


Admittedly this isn't a viable solution if you need to work with existing fasteners. Also square drive hardware can be a little harder to find, but in my experience Home Depot et al tend to have common fasteners available with square drive heads, and of course anything is available online.