Could Dart or other languages in the browser solve this problem of not being able to do crypto properly in the browser, or is this a more fundamental problem of having crypto inside a browser in the first place?
It's not language-specific. App stores already work this way (including the Chrome web store), so a deployment mechanism is there; it's just not used as often as it should be.
It's funny, back in the day we debated ActiveX (code signing) versus sandboxes (Java and JavaScript) and the real answer is that you should have both (as Android does).
But code signing isn't magic either. You still need someone to do the auditing, and users to be careful about what they run. Since users are so trusting, I'm not sure that app stores in practice are any more secure.
It just boils down to the CA based system that is already employed by HTTPS. You must of course trust the other party and their systems, but that's the exact same with code signing. To identify the other party, code signing uses a pre-shared secret (e.g. your OS comes pre-installed with their public key). In HTTPS you identify the other party by them possessing the right certificate, that's signed by a trusted intermediary. There's no really better way to do it, bar quantum crypto and HTTPS+DNSSEC, which is sadly not widely supported.
Also, in theory, an independent auditor could rebuild the app from source and verify that the bits match, and do an independent audit of the source. So signed apps are more verifiable.
The trusted base for a signed app is (browser + app + app signer), not (browser + app + server), where the server might be a virtual machine in the cloud and you need to trust the virtual machine host provider too.
This doesn't matter most of the time because you have to trust the server anyway, but in the case of someone wanting to encrypt something on the client that cannot be decrypted on the server, it does matter. Encrypting on the client is mostly pointless unless the client code is independent of the server, which requires it to be independently signed.
There is still a cert chain of course, but it's a different one where the developer's private key doesn't get uploaded to the server.
Signed apps are fundamentally not how the web works. The argument here is that the web is basically broken for client-side encryption and the app store model is better for things like secure email or a bitcoin wallet.
But since most app store apps aren't open source and the open source ones aren't independently audited in practice, it's not clear it's a practical difference.