Implementing required status checks with branch protection using webhooks?

For one project of mine, I want to implement a required status check as used with the branch protection feature. Unfortunately, none of the GitHub Apps (free or paid) are suitable for my needs, so I need to implement my own required status check.

Is it possible to implement a required status check using GitHub’s Webhooks? I cannot use GitHub Apps as they are hosted at GitHub (and I do not want to disclose private keys etc.) as I understand the environment.

From my understanding, GitHub Webhooks are only informational, and do not affect/bias GitHub’s process. Am I right? How to implement an own/private required status check?

A service triggered by a webhook can use the status API and send back a status. Having said that, GitHub Apps can be hosted anywhere, they are actually not hosted by GitHub.

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