I just watched the above video again, this time paying more attentions. I'd like to add some tips that might help others following along themselves, and perhaps avoid at least some of the long list of mistakes on display there.
-- As convenient as spray adhesives might, um, seem, they are undoubtedly the wrong thing to use in this application. For a project like this, I'd dive into my little garage utility fridge for some 3M yellow contact cement in a small can (not spray), or use some yellow 3M super weatherstrip adhesive in a tube. Both of these are solvent-based. I use the liquid in a small spray gun for bigger projects, not because it covers a larger area but because it tacks pretty quickly and we want that all even when we assemble. Or maybe buy a can of 3M #90 or their headliner spray adhesive if I know I'll use most of it on one larger project. For this armrest project, I'd use a small acid brush for the glue, and carefully detail the glue into the groove in the face of the armrest pad, and on the matching stripe inside where the seam is sewn in the new cover. I'd do a dry trial fit first of course, and make match-marks on the cover so you know where it's supposed to end up.
Follow The Directions On The Adhesive Container for application and particularly for tack time. And for clamp and dry times. Grab some metal binder clamps from your desk drawer or the office supply store, and some tongue depressors or even popsicle sticks to use as small spreaders. I use the large and medium binder clamps for this, plus I have a collection of regular craft spring clamps that get pressed into service. On your dry fit, make match marks inside the cover on that seam, so you know where 'snug' is and also where it's centered to allow the correct overlap at the ends. I make V arrows with a sharpie marker for this.
Then, follow those instructions for applying the glue to both surfaces, just a narrow stripe in the groove. and another on the inside of that seam between your marks. Let the glue set and tack per the instructions. Then carefully place one end of the cover in the groove, match mark set up where you want it, fold the cover over the edge with no glue, and set a medium clamp there to hold it. Place the cover progressively, pulling gently on the cover to make sure there are no wrinkles or bunches but no tighter. Then a medium clamp at the other end. Now go back with larger clamps and your spreader, and use them to clamp the length of the seam evenly. The spreader pieces avoid having the clamps dig in or stretch the cover, so you get a nice even appearance when everything sets up. I usually leave stuff overnight once I'm happy with the clamping and the tension on the cover, and no wrinkles or bunching.
The next day, the clamps come off and you get to judge your first-step results. If you aren't happy, you can carefully peel then roll that seam off with the round shaft of a screwdriver. Another acid brush wetted with a little acetone or even a spritz of red-can brake cleaner is enough to soften yesterday's glue so you can try again. Be aware that the solvent will trash the color on the finished face of the cover.
The outer edges of the cover get glued the same way, after another trial fitment with the clamps and match marks again. The glue goes ONLY around the inside, where the cover folds around. No Glue On The Outer Face. I use blue painters tape to mask and make extra sure on the support, and follow my guidelines and the matchmarks on the inside of the cover material. Apply your adhesive with another sacrificial acid brush. Following the directions for the adhesive, let it tack some before assembling anything. Then, starting in the middle of the longest edge, carefully line up those match marks, gently pulling the cover then clamping it with spreader pieces to prevent dents and such in the cover. Then the section on one side of the first, then the section on the other side, pulling the cover just snug enough to avoid the wrinkles and bubbles before clamping. Work your way all the way around after starting at the center of each section, so any errors hide at the ends and fold out of sight.
You may find that some carefully cut wedges in the cover will make both shaping and gluing the cover a lot easier without bunching. You have your guidelines for the edges plus your match marks, so don't cut within maybe 1/4" or so and you can avoid having the cuts exposed.
Here's a picture of a door panel glue-up for one of the summer cars done in 2017. Notice the way the clamps and wood pieces are used. The Pilot armrest is a mini version of this kind of panel project; same theory and method but smaller and with with fewer clamps.
That's actually the top edge of the door card with all the clamps, and you can see the edges of some paint stir sticks I used for clamp spreaders between the clamps and the leather.
More hints --
-- You can use wax paper as a pull/slip sheet between pieces as you line them up already glued and tacky. Get your alignment and tension exactly as you want, then gently pull the wax paper out and allow the tack faces to join.
-- You can see the small metal 'acid brush' bottom left in the picture. I buy 100 at a time in bags from the Harbor Freight store. I do add an extra pinch to help keep the bristles in the tube handle, and trim the bristles to desired length with those handy scissors on the bottom right in the picture.
-- You can also see where the original now-brown adhesive was applied with a pretty broad brush in some places. I'm not nearly as fast as the women (yes...) who originally built these in Stuttgart 35+ years ago. That old adhesive was pretty crispy by 2017 even with careful storage. Fortunately the solvent in the more-modern glue did a nice job rejuvenating the original glue and bonded nicely. The original leather cover was very carefully removed and treated to softener/conditioner front and back before being reassembled on the door card. Looking at it today, the only evidence that this was ever worked on is the still-supple leather. No glue marks or pen marks on them. Some of this stuff is critical in collector markets, so use the same level of care when working on anything.
-- Maybe to the point, anything that includes statements like "that's good enough" or "nobody will ever notice or care" is rationalizing half-fast work. Do It Like You Mean It, given the opportunity.