LFE is not a simple low-pass signal taken from the other channels. LFE is a discrete track on most n.1 mixes. The processor logic (bass management) may combine it with low-pass signals from other channels (that are set to "small") and send them all to the sub via the LFE port. Or if there is no sub will try to add the LFE signal to any other channel set to "large".
The point is that there's a lot going on the processor logic, and to send (some of) the resulting channel mixes to the sub via the speaker level outputs is asking the sub to re-process what the processor already did. The result will probably not be accurate.
On two-channel where there is no LFE, it doesn't hurt to send speaker level to the sub. In all other cases use the LFE connector.
BTW, common wisdom is that the crossover of a speaker set to "small" should be 1 octave above the 3dB-down point. E.g. if the bass of the front pair rolls off (-3dB) at 40 Hz, set the front crossover at 80Hz. (This is not a critical setting so close is close enough.)
No matter where you go. ... There you are.