I don't think standalone push is made to write music in standalone mode. The way I see it you import your live project to it with the idea to carry only the push in the stage without having to hook it up to a laptop.

Yes, it seems that a copy of Ableton Live running on a desktop is how you access the files on the Push 3 standalone via Wi-Fi. It appears like a folder in the browser.

Get an iPad. You can do all this and a ton more.

Which app would you suggest ?

Well, Logic Pro is coming out on it later today, but apart from that, it massively depends on what you want to use it for and how you want to use it.

Though I own an iPad and am waiting excited for logic - I will say that producers would prefer to have some kind of hardware to go with their software, so the push 3 isn’t really a competitor to the iPad it’s an accessory

You could get a Novation Launchpad and an M1 / M2 iPad pro for less than this, have a tactile control surface, be able to use AUv3 plugins and instruments, and not be restricted on what host you use - this apparently won't even let you use VSTs.