Even if that's how bitcoin really works, they couldn't do the episode that way Because Plot.

If it wasn't really lost, then it's not Sheldon's fault, etc. etc.

And hence, no jokes, and no episode.

I think the writers just wanted to use bitcoins in an episode and so they did.

@svetiev said:

I was under the impression that the Bitcoin ledger is updated and distributed online, so why wasn't the information about their Bitcoin recorded on that ledger and why was erasing it such a disastrous action?

I haven't watched the episode, I just read a recap so I may have missed the explanation in the story

I don't think the show staff has bitcoin experts on payroll, they just wanted to include a current topic of interest on the show.

@svetiev said:

I was under the impression that the Bitcoin ledger is updated and distributed online, so why wasn't the information about their Bitcoin recorded on that ledger and why was erasing it such a disastrous action?

I haven't watched the episode, I just read a recap so I may have missed the explanation in the story

From 6/12/18...

Three men charged in $2M cryptocurrency kidnapping and robbery

From 9/6/18 (NYDailyNews)...

“A New Jersey man who orchestrated the kidnapping of his pal so he could steal $1.8 million in cryptocurrency pleaded guilty Wednesday in exchange for a decade behind bars.”

Difference between art and real life. In real life Sheldon would be looking for a t-shirt to fit over his orange jumpsuit.

I doubt leonard and the others would press charges. And what would be the point, anyway? It's not like Sheldon could afford to repay them.

Plus, Sheldon didn't maliciously steal them. He basically just lost them.