Sunbird is closed source so you just have to take their word for it when they say they don't store messages or credentials. How the fuck could you know if they're lying or not? You can't because it's closed source.
As much as I have issues with the similar Beeper, at least Beeper is open sourcing their bridges.
That statement is pretty stupid in general. But for server side software, open source doesn't help much. Even if you can look at the source, you still need to trust them that that's what they are running on their servers.
I am often able to reach of level of trust to believe a company is not straight up lying about the code they are running on their servers.
I am not often able to reach a level of trust to believe a "trust me bro" from a company (especially if that statement is not qualified in a meaningful way).
Open source is important for services with end-to-end encryption, because you can make sure the client actually encrypts the outgoing data, is not sending your private key somewhere, and won't break that security at some point in the future.
Of course this particular service cannot even have end to end encryption in the first place.
For a bank or any system you would not have control over anyway, it does not have to be open, only the client software you run on your computer should be.
But messaging, document editing (like Google Docs), etc. are personal tasks that could be done via a local program, so a remote program should be give you freedom from it's provider.
In other words: "Some of the messaging community believes that software that can be controled by the user and is clear how it works is doing what the user wants it to. It is our view that it is not."
They are just like the rest of big companies. Remember when Facebook was a privacy respecting and friendly alternative for MySpace? Or Apple for IBM? Or Google for other search engines?