Investigate whether we can eliminate the SSE logic from DevTools app and DevTools server

[kenzieschmoll]

The SSE connection logic (https://github.com/flutter/devtools/blob/master/packages/devtools_shared/lib/src/service/service.dart/#L97) is only ever hit when debugging a Web app in g3. If this is truly a g3-only need, then we should try to remove the dependency on this logic in our external source code and move it to a g3-only location.

This will reduce the number of dependencies we have to take on in the DevTools packages.

There are some SSE endpoint on the DevTools server in DDS that are confusing because they are not hit 99% of the time (non google3 connections). This needs to be investigated further to see if we can really remove the SSE logic from DevTools.

[DanTup]

The SSE connection logic (https://github.com/flutter/devtools/blob/master/packages/devtools_shared/lib/src/service/service.dart/#L97) is only ever hit when debugging a Web app in g

I don't know if it's the same code, but I definitely still see SSE connections (externally) here:

I was looking because the per-host connection limit can cause a bad experience in VS Code (see #8298). I think eliminating SSE would make this much better because it would remove a permanent connection from each DevTools tab open inside VS Code (see my comment at #8298 (comment)).

[DanTup]

Apparently I overlooked this in your comment:

There are some SSE endpoint on the DevTools server in DDS that are confusing because they are not hit 99% of the time

I think this relates to what I posted above.