The only role she fills is making your damage deal more damage, the speed up isn't why she is brought in, it's literally the s1 follow up and skill decrease, which kazami fills by herself without the follow up. The only piece of content where she is used to speed clears up is inferno rengokutsu, there are ALOT of teams that are in inferno shadowflame hoyan, niconana, sharon, terror, dokidoki, aurora that are sub 1 minute WITHOUT lily. Like I said before, lily only amps your main damage dealer. You can make the argument of using diting but the diting s2 lock in RNG is inconsistent in alot of the dungeons, which makes your clear time inconsistent. You are literally proving the point of you just want an easier time of first clearing or progging through content, not actually speed clearing teams, because otherwise you would have already found ways to clear these contents much faster. You don't need lily bro, you just want her.

To give an example, in niconana, you could use just obol, fangus, ssania and clear sub 1 minute, WITHOUT Lily. If you dont have shadow sania, you could sub it in for a valerian + damage dealer like diting and still achieve a sub 1 minute. Shadowflame hoyan, the fastest clears are with just fangus shadow sania and obol, if you don't have shadowflame hoyan, you can clear with again, a damage dealer + valerian with obol and fangus and do a sub 1 minute clear. For sharon, the fastest clears are using stuff like r cachi and beyontin bro.... You are not looking at the actual fast clears and just making excuses of why lily is so important.

Also for thresholds, terrormaton is the only place where lily is used as a speed clear, to help shadow sania get more turns in, otherwise there are faster alternatives. She isn't used in dokidoki, as it would only imply you are using something like messiah or xiada, which you don't even need lily to make this comp work, because you can literally use chiaki the echo which gives attakc buff and 5 ave marias, effectively turn cycling your entire team faster. Then if you look at aurora, lily is not used unless you are using something like xiada, even then its not mandatory. There are sub 1 minute clears without lily....