That depends on the bank.

Some allow you to sign the papers online and only require you to authenticate yourself to make the contract binding. This authentication can be done by letter as you said, it in person in the post office or with a video call in which you show your Ausweis.

Your bank just might not support these options, but some do.

Also, it depends on country. When I was opening a bank account in Czech Republic, they required a passport and that's it. In Ireland though, you need to wait until you get registered for taxes, then you need to show the proof that you work, then an accommodation contract in order to open a bank account. I spent a month without a bank account.

Na, you can do it all online for many Banks with some of the id check methods. It's just legally preferable for Banks to have a physical signed contract as there are a lot of responsibility on both parties sides so many Banks require you to sign and send them a contract. So it's not because they technically/legaly can't, it's just because they prefer not to for legal liability reasons.

I recently opened a bank account on a major german bank (commerzbank). It took 15 minutes and I didn't have to send a single letter!

I opened an n26 account via their app in Germany. Took me 10 mins including the kyc process