I've watched a few videos of people playing and it appears (without the footage being cut) that they're just pressing a button and it restarts the whole thing instantly in order to start with a specific item. Is this a program/macro? Is there a button I can press for this?
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
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.
No controller has binding for restart.
We ran out of buttons. Maybe in an update.