Browsers are a concern for the same reason a cloud providers: you are running untrusted code in a sandboxed VM, and Spectre allows you to potentially exfiltrate data from outside the sandbox. Cloud providers are a bigger concern because they're more likely to contain interesting data* and because it's harder to exploit Spectre via Javascript than native code, but there are Spectre proof-of-concepts written in JS.

* interesting to an attacker, relative to the effort required