I've often wondered this myself. There is a manufacturer that eliminated the downtube and replaced it with a cable.

I have broken the seat tube on a frame and ridden it home that way. It was more flexy, but it was rideable.

When you ask these kinds of questions, you get an immediate kind of response exactly as you see above, usually with no empirical basis.

Based on what it felt like to ride a bike with broken seat tube, I think you'd find WAY more flex, and as the frame flexed, your seat tube and headtube angles would change. I believe you'd notice that in the way the bike handles. Would the frame eventually fail? It probably depends on the frame. If you have a chunky steel frame, it might just flex and steer differently, but hold up indefinitely.

If it's a steel bike you can afford to lose, my advice would be to try it and see. Steel fails in a slow predictable way. You'll see if it bends past its ability to recover or start to crack. Take it on an easy ride first, somewhere flat.