Again, this was designed for the average player, and I am sure my fellow repairmen will chime in with ways that they would of done it differently.
I wasn't really going to but why not
I do a lot of things differently. Just the way I like to do it. Here they are:
I don't use a razor to remove the old cork. I use either pliers (without damaging the neck at all!) and/or a scraper. I sometimes use heat/flame to make the old cork easier to remove (they vary a lot in how good they are glued).
I sometimes follow with a brass brush in my micromotor to remove all the little bits of old cork and glue if there are any that are really difficult to remove. I also use a solvent to clean it. I doubt it really matters, but I try not to touch the cleaned area with my fingers after that. Actually it probably doesn't matter at all considering how I put the glue...
I don't bother measuring so accurately with a ruler and a string etc. Most often I just put the neck on the cork and mark the width (length from neck perspective) then check by loosely wrapping it around the neck. Occasionally I use a ruler or caliper for approx length. I use a ruler to cut against for the length and sometimes for the width too. Basically whatever is fastest at the moment since the exact measurement isn't so important, it just has to be long and wide enough (while trying to keep waste to a minimum). I measure and cut more accurately for clarinet tenon corks or sax necks with the ring at the end. I cut a longer and not as sharp angle at the end, for more overlapping surface area, but I like the sharper taper for some reasons too and might try a slightly sharper angle sometimes.
Since I want the cork to be more or less cylindrical, I sand some of it before gluing. That is a long, gradual taper, towards the part that is going to be the sax end of the cork. How much depends on the shape (taper) of the neck, length of the cork, etc. Then I have a lot less to sand after the cork is glued.
I use a finger to spread the glue. This is why I said not touching the neck with my fingers is probably not important. It's probably my imagination but I think maybe "greasing" the neck accidently with any skin "grease" might be an issue, though touching the glue itself maybe won't. Anyway I don't know, just used to doing it this way.
I've tried a brush and the result was pretty much like yours. I prefer a much thinner layer of glue. Maybe that's also why it dries much faster than 10-15 minutes. Usually less than 3 minutes. I sometimes spread very fast a very thin layer on the cork, then the "real" layer. I don't think I have to do that, but cork can absorb the glue sometimes, even good cork. After I do that, or if I don't, I put the glue first on the neck. Then I spread the glue on the cork, first on the back, starting from the bevelled end side, then on the small bevel. I do this because I want to put the glue last on the part that is glued last, since it will have a few more seconds to dry.
After the cork is glued I shape it so it is more or less cylindrical. I prefer this because it gives a firm and tight seal and grip at the start but doesn't get exessively tight and hard to insert the more you push the mouthpiece on. I don't use loose sand paper to sand, only nail files and/or sand paper glued to rigid tools. I like to have more control with a rigid sanding tool. I sometimes put some cellotape at the end after the cork just in case.
Also if I'm sanding after using a knife or razor, I first move the knife/razor away from right under my hands where they can touch them and rub against them, etc.