Yeah, this is something I’ve run into as well. The site picker there doesn’t actually show all the SharePoint sites you have access to. It mostly pulls from recent or cached sites, so even if you have permission, it won’t always appear in that list.
The other issue you’re seeing—where all sites seem to show the same lists—is also related. The dialog doesn’t refresh properly when you switch sites, so it ends up showing lists from a previously selected site. It looks like duplication, but it’s really just the UI not updating.
And yes, that small dialog box is another pain. There’s no way to expand it, so you can’t clearly see long URLs.
What usually works for me is not relying on the dropdown at all. I just paste the full SharePoint site URL directly into the field and let it resolve. Once you do that, the correct lists should load.
If you’re still having trouble, another approach is to connect to the SharePoint site first from a canvas app or flow (where the picker behaves better), and then reuse that connection inside your solution.
Overall, it’s more of a limitation in how the SharePoint connector behaves within solutions rather than anything wrong on your side.