Right now, most or all runs take Violet Center after Falkner and then do the double deposit after Abra-warping to Violet (I believe Underscore might have been skipping Violet if he had healing items leftover). Skipping the heal after Falkner is risky if you are not picking up the Route 32 Super Potion (whereas the Super Potion can be safely skipped if you heal at Violet and keep your leftover healing items). The second heal appeared to be unfortunately necessary because of Headbutt PP wasted in the Goldenrod fights, and the fact that you are normally low HP after Whitney and running low on healing items.
Now I am thinking that you should usually be able to double deposit after Falkner (if you have a slave) and still be able to stretch out Headbutt PP through Chuck's gym without having to take a center or pick up an Ether. This should only present an issue if you are very low on healing items and/or Headbutt PP, but with some new ideas, these risks might be mitigated as well.
***EDIT: From initial testing, it looks like entering Violet center just once (post-Falkner) is a free-ish ~10 second save most of the time. Taking the Nurse heal + single depositing the egg is comparable timewise to double depositing + switching Pokémon, so going slaveless up to Falkner is no big deal. Super Potioning at Sudowoodo instead of a second heal in Violet center means 13+ seconds saved as a base. You may give back some SE text and flinch chances to make fewer Headbutts work, but this should only be a few seconds.
This may make Early Underground unworkable most of the time, and early Ping and/or early Strength may come into play on low Headbutt counts, but Headbutt PP in general hasn't looked too concerning (it's tight, but comes out fine with good planning). The main question is whether burning the extra Super Potion puts you at major risk of running out of healing items without picking up another one. With the new Ecruteak strats, testing so far has only had this issue come up on runs that don't hit any spinners (excepting Don) and don't fight Ralph (i.e. fewer Azalea Mart healing items + fighting Sudowoodo for exp). And even in those situations, it's not clear how dangerous it is to skip a healing item pickup.
Would say the jury's still out probably, but it's looking pretty good right now, barring any oversights on my part.
Setting up on Morty's Haunter
With trainer AI findings, Exarion had the thought to kill Morty's Gastly on turn 1 and instead setting up on the L21 Haunter to reduce chances of seeing Curse (instead of a 50% chance to be greatly encouraged for 2 turns on Gastly, it is treated neutrally for 2 turns on Haunter). If Toto doesn't have his Przcureberry, this is an even stronger consideration since Haunter doesn't have Lick. Also note that Hypnosis (and status moves) are apparently nerfed in Gen 2 for NPCs (45% (?) accurate instead of 60% accurate). These things should mean considerably less damage taken, and the possibility to conserve a healing item if cutting it close.
Sage Ping before Kimono Girls
When looking for ways to conserve Headbutt PP, I came up the idea of possibly fighting Ping before the Kimono Girls (not possible on Crystal since the gym is empty if memory serves). You can 1-hit the Gastlies with Bite and get 1625 experience, or an extra level for each Kimono Girl essentially (1 @ L23, 2 @ L24, 2 @ L25). You waste about 7 seconds of movement and some Super Effective text, but get some benefits that may make it worth it.
At L25, Vaporeon becomes a probable 2-hit with Headbutt on all attack DVs (extremely likely on 12+ DV), and you can speedtie Espeon with 6-7 DV + outspeed with 8-10 DV (L24 is 9-10 tie, 11-14 outspeed; L23 you can outspeed with 15). Also, Umbreon is always 3-hit by Headbutt with bad attack, and can also die often to HB + HB + Scratch on good attack. At L24, you can also choose to Water Gun Flareon twice, if Headbutt PP is super tight.
The time lost from early Ping can be saved during the Kimono fights and the extra experience should allow for more efficient movement in the Kimono theater. They also can save health and Headbutt PP which could make skipping the center heal [more] viable, saving an appreciable amount of time. Haven't quantified it, and it may not always be worth it, but I'm struggling to see why this wouldn't save time in several situations.
- If you don't catch a Sentret, could it be worth catching a Pidgey (skip Kenya) or a Bellsprout just so you can double deposit after Falkner? And what to do if slaveless (healing items or no healing items)?
- There's no good menu after double depositing post-Falkner, but this is a minor thing since swapping Croconaw to the front at Sudowoodo only enables a minor timesave from selecting Squirtbottle from the menu.
- With these possibilities saving health, there may be shopping options that swap out a [Super] Potion for something else. Not giving a lot of consideration to this right now.
- Have to be conscious of Headbutt PP in the Goldenrod fights if skipping the center after the Abra warp. This could impact the decision to do Early Underground, which uses more Headbutt PP than normal strats and forces teaching Bite over Scratch or Water Gun, which limits Headbutt conservability after the fact.
- Headbutt PP will be lower and Surf PP higher (if early Ping) for Lighthouse and Chuck. Haven't thought about this part a lot, but Headbutt is nicest for Noctowl, Marill, and Poliwrath. Picking up Strength early may be the right play on very low Headbutt counts.
- Since you are using fewer Headbutts overall by not refreshing its PP, you do lose a bit of time on average from not flinching things. Can't be ignored in the overall equation.