Basically, the idea is to remove the huge GUI on the side and replace the vanilla recipe book with a custom one which gets its data from JEI. Either adds a custom button to the inventory, or replace the pop-up GUI of the recipe book itself. Not sure which one is easier.
The search bar would have the same functionality as the one from JEI, and the vanilla "only show craftable" filter button would function as expected.
If there are too many tabs, it adds an arrow to scroll down/up.
I think this is an interesting idea for integration and might work well with the new bookmark system also.
Also as someone that has looked into REI (Roughly Enough Items) and how it does 'show craftable' it does that and what's in your inventory so well and better than the Vanilla Recipe Book does things since the Recipe Book is a research system more so than a craftables of all you have in your inventory.
So it makes sense but due to what JEI and mods like it's goals are I can see that being interesting.
Also as someone that thought the Recipe Book was a good idea I'd like to see mods add more alongside it even though I don't use it I'd love it to get some support for those that do.
Seeing it for JEI for Forge environments over say Fabric (1.14 snapshot modding) or Rift (1.13 & 1.13.2 6 months+ before Forge 1.13.2 modding was a thing but a lightweight modloader so it differs on support, like Liteloader and possibly what exists to take Liteloader's place for 1.13+ since nothing has come about it since so many think Liteloader is dead) would be good like REI does. Just thought I'd point it out knowing how I've seen it done but would like to see it for JEI when I use Forge and mods in that environment over the other modloader alternatives options and concepts/features.