I checked pretty carefully by reaching in and pulling the cone outwards lightly (but with enough force to move a bit) and I don't see any detachment
Well they can be hard to find until the cone is removed.
When you push the cone in and out, do you hear a rubbing or grating sound?
The fact is that when a speaker has a buzz or rattle, that is not due to a rotted surround, or a surround that is partially or totally unglued, then the cure is always reconing.
So you need a new cone with VC, spider and surround. You also need a new dust cap and a set of factory shims to align the voice coil in the gap. You will also need loudspeaker glue.
So you need to desolder the wires of the VC from the chassis terminals and remove the cone. It is no use, so it does not matter how much damage you do.
Now put masking tape over the VC gap.
Now all the glue has to be removed from the chassis. This can be done with acetone, or alcohol and a paint scraper. Alcohol is slower but I like it better. It takes longer, about 20 min to soften the glue, but I find you get a better peal. You need to remove all the glue from where the surround and spider attaches.
Now blow out the VC gap with compressed air. Now fold a piece of masking tape back on itself so there are two sticky surfaces outwards. Now wipe it round the gap to make sure there is absolutely no dirt in the gap.
Now place the new cone in the speaker. Put the factory shims between the VC and the pole piece and leave them there.
Now glue the spider to the chassis with no stress.
Now do the same for the surround.
Now leave everything alone for at least 12 hours until the glue is fully cured.
Now remove the shims and glue in the dust cap.
Resolder the VC lead in wires to the terminals.
In 12 hours your speaker is ready for installation.
Check with BIC, for a recone kit or new driver.
You can also check with Orange County Speakers, or Simply Speakers and see if they have a recone kit.
If you can't get a new driver or recone kit, then you will have to use the driver you suggested in your original post and to the mods already outlined in the post above.