It all depends on how you define "influence." If you are hoping to choose a specialty with highest percentage of "life and death" interactions with the highest likelihood of tipping the life/death balance towards life, you're view of what it's like to be a "real doctor" is one from 30,000 feet. Even the most life and death specialties (ER, ICU, Neurosurgery, Trauma Surgery) are dominated by routine cases, less dramatic decisions, lots of charting, lab/xray checking, routine non-life saving procedures, administrative pressures, co-worker interaction, administrative interference, business realities, utterly worthless and mind-rotting government "boxes to check" (ie, "meaningful use"), medical-legal concerns, and non-medical time pressures.
Yes, in Trauma, you will have the occasional gunshot victim, brought back from a flat-line with a thoracotomy and evacuation of pericardial tamponade. But you'll have lots and lot's of "fell hit head, and slow to wake up," and "consult ORTHO," "CT normal," "transfer note to rehab," or "spleen lac, admit, non-op, watch vitals, advance diet, 'did you poop yet?'"
Yes, in Neurosurgery there will be subdural evacuations that make the comatose live, breath, walk and talk again. But there will also be lots of "back pain," "foot drop," and "brain tumor too big, sorry can't help."
Yes, in ICU you will turn around the occasional spiraling patient with sepsis and ARDS and live to see them walk out of the hospital, smiling waving and saying, "Thanks." But you'll also have the "trach, peg, transfer to floor," "call family, discuss code status," "repeat culture, repeat culture, repeat culture."
Yes, in ER, you will have the occasional airway or cardiac defibrillation maneuver that takes a dying patient from the brink. But you'll have a lot more "uri's," "toothaches," and "kidney stones."
But these things are only clips from the highlight reel of the "profession" and "career" which much of the time like something that resembles what other people call a "job." That's right. A "job." One with a "boss" you didn't choose, a paycheck that's not guaranteed, with downside liabilities, with huge responsibilities, with future uncertainties, with co-workers you like and others you don't, with "shop politics" and the same unrelenting obsession with "budgets," "profits," "productivity," and a "system" that is bigger than you and I as utterly terminable and replaceable "providers."
On the flip side, you can make a "difference" with each patient you see, no matter what you do, if you define it the right way and choose to see it. A good doctor is a good doctor. On the most basic level, being a doctor simply involves one person trying to help another. For a dermatologist, that might mean finding that melanoma right before it metastasizes. For a pathologist, it might mean seeing that early cancer that someone less trained, less diligent may not have looked hard enough to see. For an internist, it may mean convincing a family to put their demented mother in a nursing home a day before she falls and dies of a subdural, not after she falls and the decision becomes unfortunately obvious. For an Emergency Physician having an "influence" may involve being the first person to spend 60 seconds non-judgmentally convincing an alcoholic it might not be a bad idea to try rehab just one more time. Or the surgeon that decides not operating is best, saving a patient a potential surgical complication. A Pain physician may save more lives by skillfully deciding who not to prescribe pain medications to, than by any pain relieving treatments he can provide.
In our blood sucking, soul-crushing, schizophrenic, illogical, money/government/administration/insurance company/lawyer/politics-driven system that we find ourselves trying to remain "doctors" in, there still are way to have an "influence" with each patient and make a difference, no matter what specialty you are in. 99% of the time, no matter what specialty you are in, it will not take the form of heroic and skillful maneuvers, but will simply take the form of you trying to help another human being in some real, but underwhelming way that makes little if any difference to anyone but you and (sometimes, but not always) the one you've helped, or at least tried to help. Most, if not all, of this "influence" will not be detectable on patient satisfaction surveys, recognized by administrators or result in a larger paycheck. But if you can see through the smog, smoke and mirrors that the insanity of our current healthcare-less system throws up, you just might find its enough to make you feel you make a bit of a "difference" to this world in a meaningful way.
So to answer your question, "What percentage of high acuity cases do you really influence?"
The answer is "almost none," or "almost all," and it depends entirely on whether you choose to see the "Mona Lisa" or the "missing eyebrows."