When I started out going lightweight, I carefully removed all labels, thinking I was cutting weight. I stopped doing so after about a year. I discovered that I needed the size information and, especially, the fabric care information later on, and I inevitably lost the tags. This is especially a problem if you might want to sell the item or donate it to a charity thrift store later on. You of course never expect to do this when you first buy an item.
I also found that it takes a LOT of labels to register 1 gram on my scale!
Removing labels that are sewn into a seam weakens the seam. If you do remove them, just cut the label close to the seam. Otherwise you'll have to open up the seam (very carefully as to not damage the garment fabric) and re-sew. Of course you never have matching thread, so you either have an unsightly mess or have to go out and buy thread that matches the garment.
If you remove flat labels as shown on the photo of the sleeve, you'll probably have to seam seal the resulting stitching holes, which completely offsets any weight savings. You'll also considerably reduce the resale value of the item. Remember that most people regard these tags as status symbols!
I do remove tags at the back of the neck that bother me (I wear the shirt a couple of times first, since not all tags are a problem). Fortunately, a lot of manufacturers now print the info on the inside of the fabric of the shirt!
I also remove (by cutting about 1/8" from the seam) the "Do Not Remove Under Penalty of Law" tags on sleeping bags. Those, of course, weigh more and are a nuisance. The warning on the tags is for the retailer, not the consumer!
I know what you mean about being a walking advertisement. I try to avoid buying clothing with enormous logos. As long as the tags are small, as on your Patagonia jackets, nobody's going to notice them. It's not as though they scream "Patagucci" to someone 50 feet away! I have several Montbell items which have "Montbell" embroidered on them. Removing the embroidery (a rather hazardous and lengthy undertaking) would leave a far more obvious mess.