My concern was I've read something years ago that said if you don't get the glue just right the piston action of the cone would be compromised.
Voice coil. The voice coil is centered in the voice coil gap, a very narrow gap with tight clearance between the magnet and pole piece. The voice coil is held in position by the spider, the accordion looking thing you can see if you look at a driver up close. Voice coil, spider and cone all come together here.
The surround is way out at the wide end of the cone. So it has leverage and yes if you screw it up bad enough it can push the voice coil out of alignment and if it scrapes in the gap this is very bad indeed. But this will not happen unless you really screw up and glue the surround so everything is all cockeyed and off center. So just be careful and then, no worries.