How can I make an automatic cobblestone breaker?

It is impossible to break blocks like cobblestone using only redstone or pistons. However, there are several mobs such as the Wither, Ghast, and the Creeper that can break blocks. My suggestion for making a block breaking system would be to use these creatures of mass destruction.

Not possible in vanilla. For automatization you should use some mods, they are just for this.

If you want really a lot of cobblestone vanilla way, just make yourself Effi 5 diamond pickaxe and place beacon with Haste 2 , then you can break mountains in seconds :)

If you do skyblock, automatic grinding of stone is maybe useful, but can be considered as "cheating" :)

Here is a proof-of-concept video, illustrating a system using TNT. The cobble is generated and blown up by TNT; and then transported by water to (presumably) a hopper. There are other ways to achieve the desired effect -such as using creepers/withers/etc (through a monster spawner or darkened room) - which could be done, for example, by using creepers and ocelots with directing corridors, to get the creeper to where you wanted him to go.

If you're looking for an easy way, just get a whole bunch of gunpowder and make loads of TNT. Otherwise, I'd suggest trying the "creeper and ocelot" method.

You can use command blocks. Place a redstone torch under the block you want to break and a command block above. Then, the command is:

/setblock ~ ~-1 ~ minecraft:air 0 destroy

Other than that, there is no vanilla way. (this is also not very vanilla)

To pick it up by a hopper place the torch on a wall under it and place the hopper at least 2 blocks under the cobble to make sure that you don't power the hopper

Tnt is a good way of breaking blocks, since they explode. Beware that they can also destroy the blocks too, so it isn't a good way of collecting either.

One way to do this wold be to have an observer block facing down at the cobblestone which triggers a piston to push the cobblestone when it appears, you have pistons further down that push these lines of cobblestone into a big stockpile, there you can mine it or blow it up.

The "legit" vanilla way is using TNT. Normally, with a single TNT breaking about 15-20 blocks and costing 4 sand and 5 gunpowder that's not a very viable approach, but there is a way to improve that by a lot: blast chambers.

Blocks while moved by pistons have blast resistance of 0, technically a bug but in practice this issue was addressed by Mojang in snapshots of 1.14, tweaked some, reverted, tweaked in a different way, then left in current state, pending a "proper fix" in indeterminate, distant future. Yes, it can be removed, but it's unlikely to happen anytime soon, and meanwhile blast chambers work just fine.

The technical community is in an 'arms race' to produce a blast chamber that breaks most blocks per one TNT explosion. Here's a popular design by Ilmango that breaks 240 blocks per one TNT.

There exist ones that double or triple that, already.

And if you don't mind exploiting the TNT duplication glitch (most technical players don't), you can get by with a much simpler designs that don't time the motion with the blast perfectly, just happily drop as much TNT as you need, or still utilize the perfect timing and achieve insane drop rates - for example this cobble generator produces 120,000 blocks per hour (over a chest per minute):