People would still farm without subsidies.

First: they are growing some crops now already that are not subsidised.

Second: a general drop in subsidies will mostly just result in lower land rents. The reward for farm labour will stay roughly the same, because it's mostly set by the general equilibrium in the wider labour market.

Beyond patronage, one of the primary goals of ag subsidies is to prevent farm land from being converted to non-farmland. This effectively increases the ability to comparatively quickly put more crop acres there if for some reason (e.g., weather, war, etc) we need to increase supply. Without subsidies it's likely that the lower land rent would cause more fields to be converted to other purposes and we'd run this a bit leaner (i.e., more in accordance to the market on some shorter time horizon). There's also always an element of keeping more expensive ag in country than relying solely on imports.

I don't know how strong this effect would be, but I've heard these arguments a lot. A good deal of early EU politics were measures to make sure that liberalized trade didn't just result in the French agricultural industry vanishing.

Yes, that argument is common, but it doesn't mean that it holds water.

If you actually wanted to ensure a steady food supply, you'd use subsidies that directly target that.

Eg instead of subsidising actual production of a few specific crops, you'd have subsidies that reward being able to produce eg lots of calories in a field on short notice. (And every year, you'd challenge a random selection of recipients to produce the promised calories on short notice. Of course, if you use that land for already for actual production, that's fine.)

You'd also reward people to stockpile lots of canned food, I guess?

> A good deal of early EU politics were measures to make sure that liberalized trade didn't just result in the French agricultural industry vanishing.

Yes, but that's more of a function of the power of the agricultural lobby, than any rational policy.