before I get my coat...
Read Peterson and Hill. But in the mean time...
You are playing all the right notes but not necessarily in the right order. Also what you think are constants aren't and what you think are variables are but not necessarily in the way you assume. You also think the physical dimensions of the engine define everything.. er not quite.
All, engines work best if the input is cold and the output to the "flywheel" for the want of a better word is hot.
For the moment forget density. An air breathing engine wants oxygen. It just so happens there is relatively speaking more of it at sea level where air is most dense. If an engine needs oxygen and the designer can help it find oxygen in dense and thin air, believe me the oxygen goes in all right and the effective mouth size gulping it is bigger or smaller than the hole at the front. A carefully designed inducer and diffuser does this in a jet but is dam' more difficult to persuade an unducted fan or prop.
A prop just wants a working fluid and it could be five-fifths nitrogen for all it cares. But it is a profligate user of air, neither inducing it neatly through the disk nor giving two figs where it goes afterwards.
A jet operates by thrust created by the compressor and partially used by the turbine that turns the compressor but if the compressor overpowers the turbine so to speak we have thrust for flight. Otherwise it chokes and stalls.
A choked airflow can always be unchoked without loss of pressure by heating it further. Up goes the temperature, up goes the speed of sound, away goes the choking. If a prop ever, god forbid, got choked there is very little recoverable work because temperature in and out are much the same and I forbid you to throw petrol or kerosine out of the window and set fire to it to heat up the air and unchoke the flow.
On the other hand if you can gently induce and contain the airflow into a tin can I don't mind the compressor compressing the air so that by the fourth or fifth stage it chokes. Firstly it has got a lot hotter so it can do work... heat and work are interchangeable. Secondly I can unchoke it at will by a controlled burn of kerosine in an enclosed space. Heat the air, increase the speed of sound and choking is gone. So the burner really unchokes the choked compressor to give the turbine its best fighting chance to turn the compressor and some extra fuel is chucked in to cover the losses. Thirdly, choked the compressor might be but that only sets a limit on the amount of air the engine can ingest which is determined by the effective intake area which is not the same by any means as the physical dimensions. Come with me and I will show you that a running engine ingests air not from dead ahead only but from all over the shop. True in some flight regimes the effective area is smaller than the physical size by tape measure but the overwhelming majority of subsonic and trans-sonic machinery sucks in more than outward appearances suggest.
So the jet engine we know and love so well really hands the designer several tools to take it past the obstacles provided by nature. Now... TSFC.
TSFC? It will be a lower value at sea level than at altitude. But don't be fooled. Fuel consumption is always better where oxygen is aplenty but you are missing the point that it is thrust specific. How much thrust you need muchly depends on lift/drag ratio as well as weight of course. Poor L/D occurs down low and slow and completely swamps the oxygen rich fuel consumption benefits displayed on the data sheet. Come on! You know a 747 burns fuel at a rate of thirty tons an hour on take-off but half an hour later she's guzzling at a rate of ten tons an hour. All things equal lift is proportional to air density but the square of velocity. Retract those flaps and wheels, get her up where the air is cold and speed her along. Wonderful L/D ratio. Pity the TSFC doubles but thin or not the air will be drawn in because the inlet designer did a good job.
So TSFC might look good low and slow but my god you need lots of thrust to stay in the air. High on the fly TSFC may look twice as "bad" but hey L/D is not say 7:1 or worse but 17:1 and even better.
Oh I'm rambling b*gger it.