How do I do a quick restart?

It's not pressing a button, it's holding a button (not tapping).

When you hold r for a couple of seconds, it will restart your run. (However this will not work for Eden, since it uses tokens to check if you may play him)

You can also hold the left Ctrl for a few seconds to drop your trinket(s) and card(s) / pill(s) (You may want to drop a certain trinket before going to a certain place

Drop The left hand before going to the chest

Controller

For controllers the controls are different (for example: Holding RT (right trigger) will drop the trinkets and pills / cards.

And from seeing the Reddit thread about controller support:

Unfortunately, I don't think the DS4 has a restart button enabled by default. So if I want to restart, I need to switch over to the keyboard for a second. Other than that, loving my DS4.

And

No controller has binding for restart.

Statement of developer on reddit:

We ran out of buttons. Maybe in an update.

hold both R3 and L3 for a few seconds