Let's say I want to plant two tomato plants every 12 inches, or perhaps you could say one plant every 6 inches. All the plants would be supported by a string trellis, and I would be pruning to a single stem. Would this be feasible, or would the roots get too crowded?
I think that 6 inches apart is a lot closer than I've seen recommended. I've been planting 24 inches apart, and the plants get very crowded at that spacing. You need good air circulation to keep down disease problems.
You would end up with a low ratio of fruit to stem that way, and generally it's a good practice to leave at least one 2nd stem in case one of them gets damaged.
I agree that 2' spacing is better though it also depends on how large you expect them to get, whether determinate vs indeterminate.
It also depends on the orientation of the row. With them close together you'd want the row running north-south as much as possible or else they end up shading each other too much, although more shade is good if growing in a very hot environment.
Beyond what I've mentioned, given that you're using a trellis and willing to trim them, I'd go for closer to 1' apart.
Roots are fine but if you get a disease you are in trouble and overcrowding contributes to many things. 12 inches is less than recommended but that's what I do and results are more tomatoes than my family can eat.