Telegram is the only massively popular messaging service that allows everyone to make sure that all of its apps indeed use the same open source code that is published on Github.
Not true. Signal has a very similar client verification process to Telegram's, described here. The lack of an iOS reproducible build is an Apple limitation / nuisance.
It’s very complicated, the 2nd jailbroken device is necessary because there’s no other way to download the .ipa, but even if you manage to do that and bit-for-bit reproduce the .ipa you downloaded from source, there’s no way to know if the App Store is sending every user the same .ipa or if your other, non-jailbroken iPhone downloaded a backdoored one.
Telegram docs even acknowledge these limitations.
Ultimately, this client verification is not the selling point Telegram's founder makes it sound like, since most messages are not E2EE and the server code is closed.