LFE vs High-level subwoofer connection, for FIDELITY

The problem with a high-level connection to the amp or main speaker terminals is that there’s no way to properly integrate/balance the sub with the main speaker.  So-called double-bass is completely unpredictable.

OTOH, the success of the low-level connections will depend on what precedes them in the signal path but, at least, one can insert a proper crossover.

Of course, this comes from someone who believes there is no valid reason "get tube or amp sound into your bass region."  

@kr4 

The problem with a high-level connection to the amp or main speaker terminals is that there’s no way to properly integrate/balance the sub with the main speaker.  So-called double-bass is completely unpredictable.

Just run room correction. It sees it like any speaker and knocks down peaks like usual.

Does one add to "lag time" and does this even matter?

From the mains preamp, a dedicated sub runs through its amp then its speaker, just like the main channels.

On the speaker tap, the sub's signal runs from the preamp through the mains amp to the speakers, then has to go to the sub amp, so the signal arrives a "moment" later than the mains.

If you want the same dynamic signal that's coming out of your amp to be shared by both speakers and subwoofer then there's no question that the high level input is the way to go.

I'd also suggest stacking the subs but that's another topic.

Per bartsw:  Just run room correction. It sees it like any speaker and knocks down peaks like usual.

Sure. First, I suspect that this will not appeal to the OP.  Second, it's easy but can't correct main and sub separately in the overlap region where it needs to (no crossover).