[7.x] [Core] Remove public context (#88448)

[mshustov]

Backports the following commits to 7.x:

• [Core] Remove public context ([Core] Remove public context #88448)

[kibanamachine]

💚 Build Succeeded

• continuous-integration/kibana-ci/pull-request
• Commit: 91334d0

Metrics [docs]

Module Count

Fewer modules leads to a faster build time

id	before	after	diff
core	449	447	-2

Async chunks

Total size of all lazy-loaded chunks that will be downloaded as the user navigates the app

id	before	after	diff
graph	1.3MB	1.3MB	-10.0B

Page load bundle

Size of the bundles that are downloaded on every page load. Target size is below 100kb

id	before	after	diff
core	565.3KB	563.0KB	-2.3KB

To update your PR or re-run it, just comment with:
@elasticmachine merge upstream