Any thoughts on why our alfredo didn't emulsify properly?

Tonite we tried making pasta alfredo for the first time. We used the (recently trending?) bowl method so we

1. Cooked the pasta (about ~8oz of cellentani/cavatappi)
2. Scooped it all directly from the water into a big metal bowl
3. Added about ~3tbsp butter, cut into separate pads
4. Stir/shook it all til the butter melted (I thought we did it pretty vigorously!)
5. Slowly added in ~1c of parmesan that we'd grated ourselves while continuing to stir & shake
6. Noticed that the cheese was becoming a bit stuck together/gloopy despite adding it slowly
7. Tried adding some more butter & pasta water while stirring and shaking
8. Put it back in the (empty) pasta pot to try to use some of that residual heat to melt the cheese more, still stirring/shaking

Putting it back in the warm pot did seem to help the cheese melt a bit better, but it still ended up kinda gloopy/cheese stringy rather than a creamy sauce, even after we added a bit more butter & reserved pasta water.

We still added some pepper and ate it, along with the chicken we'd made to add too, it wasn't a bad meal overall, it just wasn't the creamy sauce we were going for.

Do you think maybe we lost too much heat while waiting for the butter to melt? Or did we need a specific kind of parm rather than just a regular one that we grated ourselves? Or our stir & shake wasn't vigorous enough despite our best efforts?

I know you can't say for certain, but any tips to help improve our chances of success next time would be so appreciated! Thanks! :)