I suspect that the idea at the moment is that units consume so much less food than population that they are, in effect, self-sufficient, and that supplying them comes from the support cost. I'm happy with this.
I can live with the food cost being represented by gold support, I suppose. Doesn't mean I like it, lol.
I do agree that supply could be traced, and breaking supply lines would then be a bad thing, but this is a problem of the whole city-based system. In a real war I'm not in enemy territory until I take a city; territory I conquer becomes mine as the front moves. In populated areas (towns, villages, or maybe just any improved tile) there might be partisans later in the game, or if the enemy were under certain civics that engendered civilian interest in grand affairs, and perhaps there would need to be a mechanic to 'put down' resistance in a tile to make it one's own.
I think borders/fronts should move with units, as you say, for the purposes of determining supply lines and road movement, and then allow enemy units to "cut off" supply lines and road movement by penetrating into that territory (thinking in terms of Battle of the Bulge here). Perhaps each unit should have a 1 tile "border claim" such that if a line is contiguous, you can move a front with them, or if it's not contiguous, you risk them getting caught behind enemy lines. In a sense this would replace culture-based borders with military unit-based borders, which I think is more realistic.
Culture still has its place, but primarily in synergy effects like research, city happiness, population loyalty, etc. Not borders.
But since cities inevitably have culture borders it seems silly that I could, under your proposed system, surround a city and yet still have it cut off my supplies.
If borders were determined by unit presence, a city's borders would exist as a function of the unit occupying it. And perhaps the border "pop" could come as a result of how long a unit has fortified there.
Then the calculation of who has cut off whose supply lines, is determined by whose units are where. If you surround a city with units and their front is contiguous with the larger scale front, obviously it's the city which is cut off and the units besieged will begin to degrade in health and happiness. If your encirclement of a city is incomplete and the AI out-encircles you, the obverse should take effect, your units are now cut off. No supply costs since supplies can't GET to the units, but inexorably the units begin to degrade in combat strength as they lose health and happiness.
A quicker solution would be if supply mostly ignored culture and relied on units; perhaps a rare use for the otherwise poor idea (in Civ) of ZOC. If the enemy could get units to break the supply lines from my cities then my units would suffer. This would automatically be related to culture because a civ has much greater mobility within its cultural borders than its enemy.
It would be simpler to let the ZOC determine borders, and charge a mobility penalty for units that are cut off.
I think it's absolutely stupid that the presence of a cathedral in some city throws glue into the treads of your tanks such that they can't use a freakin' road. But if those tanks are cut off they'd move slower simply to conserve fuel and try to avoid ambushes, etc.
It would also very nicely give rise to decorative units sitting in capitals that actually are quite good. Instead of keeping some idiot longbowman with city 1, he'd sit there and not do anything, and end up being city 3, combat 3 or something similar. That relates much more to the way things have worked, where Royal Guards or similar were taken from the best troops, even if they just sat around at home all day.
Back in the day Royal Guards NEEDED to be the best troops, because kings and queens didn't want to be the next Louis XVI and Marie Antoinette. That's another gap in Civ, is the ability of angry citizens to rise up and form, shall we say, "barbarian" units which, if they're sufficiently enraged, would put some of your back cities severely at risk unless you take countermeasures.
Although I do like the idea of archaic decorative units as a matter of tradition. Maybe as a special unit upgrade which not only changes the appearance but adds to the culture of the city where they're stationed.