A kitchen faucet is flow restricted to prevent wasting water. It would take far too long to fill a tub with one. Roman tub sets are set up for quick filling of the tub.
If you install a hand sprayer, it would be a good idea to either set the tub up so that it can be removed for hose access, or an access panel.
I don't think a lot of water drips down the hose, though I've had that thought too. You know a little bit will find it's way.