No, it’s not possible to build an app that initiates connections to Github/GitLab and have it quality for Runs on Atlassian (“RoA”)
The Runs on Atlassian eligibility rules have a comprehensive list of what disqualifies an app.
You can, however, use the Customer-managed Egress & Remotes feature to build an app that is completely Forge hosted except for egress rules that are dynamically configured and approved at runtime by the customer. While this isn’t quite RoA, it does present a data security posture that customer subsets will find strongly preferable. For example this style of integration is strongly preferred for Isolated Cloud apps compared to a fully remote app.
The decision to exclude customer-managed egress & remotes from Runs on Atlassian was/is a contentious one. You can find other discussions here in the developer community about it.
To summarise; you can’t get to “Runs on Atlassian”, but you can get to “as fully Forge hosted as possible, with data only shared with existing third-party SaaS vendors”, which has customer value but isn’t as well surfaced on the Marketplace with a special badge.
Thank you so much for this extensive answer. That fully covers my question and helps us a lot to build the right product for our existing customers and potential new users.