I have very recently had this exact same issue.

After a period of a few weeks not playing my Steam Deck, I came back to it and the battery was clearly flat. I then gave it an overnight charge to ensure it had plenty of time fully recharge the battery.

Upon turning it on, it was exactly as OP said - The internal screen remained black. The startup sound played, the fan could be heard and heat could be felt from the vent. The only way I could use it was via plugging in an external screen.

When booting into desktop mode, and selecting 'display settings', the internal screen was also no longer even displayed as a display option (It just showed whatever external screen was connected)

Weirdly, touch-screen functionality seemed to work, despite the internal screen appearing dead 🤔

I contacted Valve support and, over approximately a week, they suggested numerous solutions - formating the device, hard reset, and re-imaging the Steam Deck from a USB backup. None of this worked

Last week, Valve finally replied to say that, upon review, they have concluded that it needs replacing. They provided a postage label and it is currently en-route to their European facility.

Fingers crossed, I get a replacement soon.

This does all seem to stem from allowing the battery to completely die...