[7.17] Search session tests fail against 8.0

[rudolf]

When testing 7.17 against Elasticsearch 8.0:

git checkout no-include_type_name
TEST_ES_BRANCH=8.0.0 node scripts/functional_tests_server.js --config=x-pack/test/api_integration/config.ts
# in another terminal session
node scripts/functional_test_runner.js --config=x-pack/test/api_integration/config.ts --grep='search search'

The following tests fail (requires unskipping https://github.com/elastic/kibana/blob/no-include_type_name/x-pack/test/api_integration/apis/search/search.ts#L18):

apis search search post should return 200 with partial response if wait_for_completion_timeout is not long enough
apis search search post should retrieve results with id
should delete a search

Related #122962

[elasticmachine]

Pinging @elastic/kibana-app-services (Team:AppServicesSv)