I think if you are replacing cloth without a lot of experience there are lots of videos out there so I'd watch a few, or even as many as you can tolerate, and take some tips from each.. some cloths have a lot more stretch than others do.
the way he gathers the cloth for the side pockets , wellit seems the cloth he is using has a lot of stretch so not having any issues. Since not all have that much stretch so it might mean handling the cloth a bit differently..
similar near the corner pockets, when doing the rails.. some videos show to tension the cloth on a bit of an angle leading up to the corner pockets, to help gather the cloth more neatly when you get to stretching it over when you do get near the pocket,, It may help to keep it a bit more neat and tidy, some videos show a bit of technique to that and to someone first trying, this may not be immediately obvious.
also a harmless thing you can try , if you like the idea, is to take a piece of chalkboard chalk and you can make grid lines that are parallel to the warp and weft of the cloth, then as you stretch you can see better if the stretch you apply is tension the cloth unevenly. If its a felt, and not woven, then maybe just lines that are parallel prior to the cloth having some tension.
In the end you can always choose to ignore your grid lines. It may help you see a bit better, I see no harm in it. chalk isn't permanent..
another small thing to note, If you want to draw a line across the table, I think it looks slightly tidier if you draw the line before installing the cushions than to end it near the cushion.. your preference , its just hard to draw a straight line in the area beneath the cushion cover at the sides of the table.
I think it is just a bit easier before the cushions are in place.. and then if you like you can continue the line until it will be covered by the cushion rather than needing to end that line..
It helps to focus on the angle , speed pressure on the pen and draw in one smooth and constant motion with no stopping and starting.
That helps to get a line that isn't varying in thickness, or not quite straight.
I'd suggest you be selective about the pen you use, dont use a half dried up, or worn out one. worth testing it on scrap before you muck it up and need to look at the little mishap for a long time after.