I've been playing awhile now, and I'm largely convinced that the current rules for Medicae are poorly made and not working as intended.
First Aid allows the medic to make a medicae test with -10 per point of Crit Damage. It takes one turn and heals a fair bit. A modest medic can easily do in one turn what would take a heavily or critically wounded character days or even weeks. You probably have a 40% chance of success or better. You can do this max 1/day per person.
Extended Care allows the medic to make a +10 test on multiple people. It takes a full day, and heals a fair bit, but if you fail patients start taking highly variable amounts of damage, and heavily or critically wounded patients are likely to die.
It seems to me that you would almost never want to use extended care. It's much safer to just make a first aid check every day, and takes far less of the physician's time. Even with a high medicae skill and a single patient, you're probably about 10% likely to kill a heavily/critically damaged patient just by taking care of them, whereas leaving them alone or just giving them 5 seconds of attention per day is perfectly safe and can heal about as much.
Suggested Fixes
No possibility of damage from extended care: It's ridiculous that natural recovery with critical damage is safe, but taking your time with a moderately wounded patient has around a 1/10 chance of killing them.
Nerf First Aid (Maybe): It seems to easily and quickly produce miraculous recoveries, which renders extended care largely obsolete. You could limit it to one use per deployment/mission, rather than one per day, or don't allow its repeated use for the same old injuries. This would reflect that first aid is a quick patch for the short term problems, but some injuries require longer term attention.
If you apply " No possibility of damage from extended care " it becomes useful for critically wounded patients who might otherwise take weeks to recover. If you also "Nerf First Aid", then Extended care becomes the main source of downtime recovery for critical patients.
Note, I assume that you can't retry failed first aid checks before 24 hours, but even if your group does it differently the above should still apply.