We are trying to test an under-development Public App on a 'Managed (by us) Shopify store'. We only see an option to link the managed store but, don't see a way to install the app.
Is the only way to test an app is using a Development store?
We created our own test store with some other public apps installed, to see if our app works in harmony with other apps.
You will definitely need to use a development store. Check out the docs here.
I think there is a need to be able to test an app on a managed/live store, as the setup in such a store is much more complex than what you similate in a development store.
If you think this breaks the restrictions of the public app concept, you could restrict it, such that the app only can be tested in one managed store.
Allowing for testing in a managed/live store, could save time for the Shopify app approval team as well, as you are able to test more scenarioes etc. on managed/live shops.