The valves would prevent paste from being sucked back into the tube, but how would you force the paste out except by compressing the tube? Perhaps, if there is an unaesthetic component to your criticism of the current systen, you would approve of a hypodermic type dispenser but the tube is probably the most convenient and economic device.

maybe I'm missing something, but how else do you get toothpaste out of the tube except by squeezing it? my idea is just a modification of the conventional flexible tube. The hypodermic idea is good but it would result in a tube twice as long as usual (until the toothpaste gets used up, that is).

This is on the market as the NeatSqueeze toothpaste tube. I originally conceived the same design for dispensing contact lens solution back in 1989.

Back in 1954 I saw a roller type device designed by a design student as a toothpaste squeezer to remove the last bit of paste from the tube which seems to me to be an over-elaborate solution to a negligible problem. But a metal tube does not regain its former shape when squeezed as do plastic tubes. Tin or aluminum would do and remove the necessity of worrying about lost bits of paste. Is it the aesthetics of a squeezed tube that bothers you? I cannot think anything else would be disturbing. Flattened tubed do not bother me.