While the trust issue you raise is legitimate, it's not one we're trying to solve (or at least, not yet, and I hope not soon)
You may ask then, "Well, what's the point of this API if you're not going to solve it?"
The answer is that solution for the trust model issues are being addressed concurrently and, unquestionably, more adequately, in other W3C working groups, the WHATWG, the IETF, and other such standards bodies. For example, the development of Content Security Policy, the hopeful formation of the Sys Apps WG, offline apps that execute in their own origin, the proposed <browser> tag, the formalization of the things as basic as the web origin concept (RFC 6454), Web Intents as a service-agnostic IPC mechanism. There are a number of efforts going on to address and enhance the overall security and utility of the platform. Our WG is providing just one small part of the overall picture.
Every concern about malleability of the run-time applies equally so when you're not doing JS crypto - native or browser-mediated. Storing something in localStorage/IndexedDB? Well, now you've got an opportunity to turn transient XSS into persistent, stored XSS. Does that mean localStorage/IndexedDB are doomed to failure or fundamentally flawed for not addressing that? No. Are cookies complete and utter garbage due to all of their known issues? No (or at least, not /complete/ ;-)
I recognize that there are a variety of security decisions and trade-offs being presented in this API, decisions that will have to be made by site authors. It may be that this is completely unacceptable - and I would hope, by now, that people feeling that way would be sending mail to public-webcrypto-comments@w3.org saying that. However, for the use cases that have been presented and expressed, the trade-offs have been understood and are, thus far, acceptable, which is why this WG has continued the path its on and why the participants have, so far, believed in the utility of the API.
As far as algorithms go, our failure (thus far) to include DSA is hardly going to be the end of SSH (for which many still use DSA keys), just like our failure (thus far) to include ElGamal or MD5 are not going to mean the death of PGP/GPG. However, by not including them, it also means that there's no way to implement such applications, even if all other concerns were controlled for - and that would suggest our API is incomplete and a less-compelling alternative. Suggesting that failing to implement PKCS#1 v1.5 will finally mean the end of it is unrealistic.
So far, our stated goal has been to produce a low-level API that has applicability for a number of situations, as captured both in the use cases of the draft and in the companion use cases wiki. The core functionality starts with "Secure key store. Secure RNG". In order to support "secure key store," it's necessary to define what you can do with these "secure" keys, hence, the specification of certain operations and algorithms.
We're not (thus far) attempting to create an enveloping format - that's something that the IETF JOSE WG is doing, and that various groups and standards ranging from XML DSig to CMS to S/MIME have done or tried to do.
We're not (thus far) attempting to create the one-true-perfectly-safe-full-of-cryptographic-kittens-and-puppies "box" and "unbox" - it's interesting, unquestionably useful, but getting two crypto-geeks, let alone a hundred, to agree on what that operation is composed of is a sisyphean effort. Should it be Sals20 or AES-GCM? Why not Blowfish? Monkey knife-fighting appears more civilized then some of those crypto-political discussions - and makes more progress to boot.
By providing the low-level API, application developers can make an informed decision on what that "box" looks like to them, or relying on cryptographically skilled developers to produce libraries (ala KeyCzar, ala NaCl) to do that for them. Yes, it also means that uninformed developers can start smashing things together and leading to wonderfully painful crypto-explosions. However, you don't see WebGL being lamented for its lack of built-in VRML support (I hope...), and neither do I think this API should include the kitchen sink, bathtub, robe, and matching slippers in order to be useful and applicable for many web developers.
Yes, screwdrivers are wonderfully useful tools, and it's fairly hard to hurt yourself with one. But sometimes you really need a hammer - or a chainsaw, or a level, or a drill - in order to do the right job. This API is merely a toolbox - dangers and all - not the One True Solution to all the Bad Crypto.