Very similarly to how you did it on your iPad.
Download TestFlight on your phone, open it, and sign into iCloud. You won’t need a new invite or the link, because TestFlight betas are shared across your iCloud account, kind of like apps you bought in the App Store. You should see it there in a list (along with any other app betas you’re in, if you are), with an Install button.
I agree with your main point though. Looking at the “What’s New” list (What's New in the New Clear ✨), there’s many more omitted features than new functional ones.
I understand why this is, though: the app has been essentially rewritten from scratch to support Apple’s new frameworks and platform. Which is a good thing in the long run! But until those core functionalities are there in the new one, I think the old should be kept around.