Clarify and simplify integrations overview

[evazorro]

What does this PR do? What is the motivation?

Rewrites the overview on the integrations getting-started page to be simpler, clearer, and more direct.

I was trying out using different models to rewrite a section of the docs (as part of the team's AI Friday) and liked this rewrite quite a bit, so wanted to keep it.

What changed:

• Cut the abstract definition from the first paragraph; too wordy and abstract.
• Made the three bullets parallel; each now opens with how the integration is set up, then how it collects data, then examples.
• Moved the note to the end because it interrupted the flow.
• Framed the custom check as a fallback, not an equal option (you should only really create one if no suitable integration exists)
• Minor style fixes

Merge readiness

• Ready for merge

AI assistance

Written by Claude, approved by me

Additional notes

[github-actions]

Preview links (active after the build_preview check completes)

Modified Files

• https://docs-staging.datadoghq.com/eparish/rewrite-integrations-overview/getting_started/integrations/