Not exactly "everything cut off below 80Hz", because the high pass filter (HPF) is not a brick wall. Your speakers will still reproduce sounds below 80Hz but decreases (@ 12dB/octave in this case) from that point due to the slope of the HPF.
From a purely technical perspective, when working with (relatively) small speakers, I'd opine using the sub and HPF typically would be a better all around way to go because the sub likely has less distortion in that range than smaller speakers.
Whether or not you like how it sounds may be a matter of how well you can dial it in and or simply your preference. But, the same applies to running the speakers full range. You still have to spend time dialing it in that way too. Also, if you choose to try both ways, and I suggest you should, the settings will have to be individually dialed in as I'd expect then settings best for running HPF mains won't be the same as for full range mains (or vice versa).