I found the plastic pull tab to be useless. Peeled pretty much the whole battery out from the left side. Used a probe to "break" a few strands of glue to help. Just watch the connector and ribbon cable in the lower left corner as you pry up so you don't crimp them or damage them.
The spring contacts on the speaker bottom make re-installing a bit tricky, since they make the speaker pop up until the screws are back in. If this happens, one or more of the fingers that are supposed to be under the case will pop out. What I found worked was to put the speaker in, getting the metal fingers under the case as directed, and then hold the speaker down with while putting the two screws in, along with that triangle piece.
Bump on the comment re: no picture here shows the proper cable route. The text mentions being careful to route it under the metal finger, but no way to know where that it is from these pictures. The re-route is: around the left side of the screw socket and then the right side of the silver thing (finger) which is to the right of the battery connector.
I started with the larger part of the new antenna and had the peel and stick it a couple of times to get the smaller part to fold over, and then that part didn't quite line up with the holes for the screws. If I was to do it again, I'd start with the smaller side, screw it down without removing the backing and the fold it over the edge to make the crease. At the same time, also eyeball where that larger section fell. The original one did not land in the slight cutout on the speaker exactly, but the replacement antenna and speaker pair did line up. Then pull it apart and do it again but removing the paper backing.
I bought the cable from ifixit and did have to peel off the IR receiver, which was surprisingly challenging. I'd never come across the ribbon cable clamp connection before and so didn't know how to open it, or how much of the new cable needed to go in. I'd also suggest marking the location of the cable that attaches to case bottom, before you pull it off. A few pencil marks would work. (That's probably a step 8 add)