I mildly rebranded the terms I used in the first draft of my suggestion based on your feedback but so far made no functional changes to it.
The former version had 24 / 240 / 2400 base hours in a Consequence (1 / 10 / 100 days). I scaled that down to 20 / 200 / 2000 Shifts with 20 Shifts of recovery per day at base rate (which remains 1 / 10 / 100 days). Each Shift in a Consequence corresponds to 24 / 20 = 1.2 hours of healing at base rate, which ultimately is close enough to approximate an Hour still. You also expressed doubts about using percentages. I renamed them into Healing Boosts to avoid confusion but no functional change otherwise (1 Healing Boost / 20 base Shifts per day = 5% still). I hope the formula above looks as clean and tidy to me, as it does to you.
==================================================================
A mock up of how the suggested ruleset would see Hazou's circumstances :
TN : 30 (Good) -> Burns are quite common a ninja injury, even more so in the land of Fire than elsewhere, but these are third degree burns.
Shifts : 95 (Fireball) / 35 (Physique) * 2000 (Severe) = 5428 -> Ouch. Guess that's a big reason to level Physique!
5428 / 20 = 271.4 days to heal at base rate. In real life, severe third degree burns take around a year to heal, so that seems reasonable.
But, we have Noburi! Well, probably Tsunade would be best to prepare a Treatment Plan here, but he is not bad at all! This is high effort and matters so he takes +3 TL steps and burns FP to get +6 on the dice, also invoking "I Will Be The Next Tsunade" and "Team Uplift" to do so.
After he makes the Treatment Plan, he will spare "An Hour" a day giving Hazou MedNin attention, the base TL step for the task so rolling without a prep modifier. No invokes or rerolls either. Hazou will only perform minimal physical activities and Rest until the Consequence is gone.
Noburi (Medical Knowledge) : 50 (Base) + 6 (I Will Be The Next Tsunade) + 6 (Team Uplift) + 18 (+3 AB from Preparation) + 6 (Dice) = 86
Wound TN : 30 -> 19 Shifts, so +19 Healing Boosts while Hazou is following the Treatment Plan!
Noburi (Medical Ninjutsu) : 45 (Base) + 0 (Dice) = 45
Hazou (Physique TN) : 35 -> 4 Shifts, so +8 Healing Boosts extra on the average day Hazou is given Noburi's mednin attention.
20 (Base) + 19 (MedKnow) + 8 (MedNin) = 47 * 2 (At Rest) = 94 Daily Healing Rate
Hazou would heal an average of 94 Shifts from his Severe Consequence per day. It would take 5428 / 94 = 57.7 Rest days to recover, or 115.5 if he took no Rest, 139.2 days if he also didn't get mednin attention but followed the (S-rank) Treatment Plan. 271.4 days if he didn't do even that.
==================================================================
As for the mechanical differences themselves :
First, my suggestion integrates the "daily recovery" from Physique under Paper's system directly into the formula that determines the amount of Shifts in a Consequence, meaning it stacks multiplicatively with Healing and Rest, instead of every variable being additive (if healing and rest bolster natural healing, a healthier body should improve the recovery rate). Perhaps more crucially, Physique no longer has to be rolled at all, let alone daily, as a result of the change, removing the need for one of the checks.
Next, I made interactions between MedNin and MedKnow much clearer. MedKnow makes a Treatment Plan, which accelerates natural healing and is long term. It does not require daily rerolls either, as it is a diagnosis and a long term prescription for whatever ills plague the patient. A Mednin, meanwhile, needs to know that Treatment Plan to work their magic, because they need to know what to fix. They can come up with it themselves if they are experienced in MedKnow, on the spot if need be by moving up the time ladder (Half a Minute for -3 AB, 10 Minutes normally, A Few Hours for +3 AB), or they could follow the Treatment Plan made by a more experienced medic that shared it with them. This means that Tsunade can now make Treatment Plans and share them with lesser mednin who actually take care of the patients, quickly moving on to diagnose the next and the next and the next. This is simulationist - but impossible under Paper's ruleset.
The MedNin/MedKnow roll changes also resolve many miscellaneous issues. For example, under Paper's system, the MedNin roll does not matter at all when it comes to the degree to which someone is harmed if the medic fails their MedKnow check. This is no longer the case. Said ruleset also features an exceedingly steep failure-threshold-modifier for the recovery rate, which makes the aid of genin mednin exceedingly bad and outright harmful rather than helpful. To illustrate, rolling 29 vs TN 30 on either the MedNin or the MedKnow check while administering care fully blocks 20 daily Shifts from Rest, which usually make up the bulk of the healing in that ruleset, meanwhile rolling 30 (1 higher) does not inhibit it at all and even applies some actual mednin healing on top. This issue is also no longer the case.
Further, I made Rest a 2x multiplier on total recovery, rather than an additive modifier like in Paper's version. This means that if a character is resting under mednin/medknow care, the effect of said care is doubled by the fact that they are Resting, rather than just being added on top, which makes more sense than just adding everything together.
Establishing a base daily healing rate (20) instead of Physique checks (where Physique instead reduces the amount of Shifts on a Consequence), also means the recovery rate scales the same across all manner of relative Physiques and TNs, instead of having a disproportional runaway effect due to their interactions, which is another (minor) issue fixed in the process, without sacrificing simulationism or depth.
There are a few minor differences between the two systems unlisted above, such as "Medical Care can be time-laddered up twice for Mild Consequences and once for Moderate Consequences" in Paper's, versus "Medical Care can be time-laddered up or down by three steps at most" in the suggestion, but these come down to which approach better fits the MFD reality, and I found that many treatments shown throughout the story took as little as minutes of actual work, while some took many hours, so I found the leeway there appropriate.