Reland "Fix issue where DevTools would not be immediately available when using --start-paused (#126698)" (#129368)

**Original Description:**

> Service extensions are unable to handle requests when the isolate they
were registered on is paused. The DevTools launcher logic was waiting
for some service extension invocations to complete before advertising
the already active DevTools instance, but when --start-paused was
provided these requests would never complete, preventing users from
using DevTools to resume the paused isolate.
> 
> Fixes https://github.com/flutter/flutter/issues/126691

**Additional changes in this PR:**

The failures listed in https://github.com/flutter/flutter/pull/128117
appear to be related to a shutdown race. It's possible for the test to
complete while the tool is in the process of starting and advertising
DevTools, so we need to perform a check of `_shutdown` in
`FlutterResidentDevtoolsHandler` before advertising DevTools.

Before the original fix, this check was being performed immediately
after invoking the service extensions, which creates an asynchronous gap
in execution. With #126698, the callsite of the service extensions was
moved and the `_shutdown` check wasn't, allowing for the tool to attempt
to advertise DevTools after the DevTools server had been cleaned up.

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Co-authored-by: Zachary Anderson <zanderso@users.noreply.github.com>