It's been production ready (as in fully supported) since v1. It's the first new cross-platform runtime in over a decade so breaking changes are expected, but they were very clear about where things stood and always recommended v2+ if you wanted to move with the least amount of missing APIs. With .Net Core 2.2 now, a full 5 releases in, there should be no major issue for migrating apps or building new ones.
The naming is definitely a known problem but as far as "doesn't inspire confidence" and "breaking things for no good apparent reason" - what specifically are you talking about? There's tremendous adoption with the next version of the entire .NET platform converging around Core and all the development is done out in the open with public discussions and code if you want to learn the reason why anything was done.
Is it more that things are just different from what you're used to?
The naming is definitely a known problem but as far as "doesn't inspire confidence" and "breaking things for no good apparent reason" - what specifically are you talking about? There's tremendous adoption with the next version of the entire .NET platform converging around Core and all the development is done out in the open with public discussions and code if you want to learn the reason why anything was done.
Is it more that things are just different from what you're used to?