If it is pulling the hair spray or the tape off the bed then you absolutely do have an adhesion issue- it is just between the spray/tape and the bed and not between the print and the whatever!
I hate blue tape. While it works, hair spray is much better. So I'll ignore the tape for this thread.
There are many ways that you could be doing the hairspray incorrectly, that would result in it not sticking correctly. Here is a rough list of things to look out for:
1. Not just any hairspray will work. I use Garnier Fructis #5. I used to use Aqua Net, but after doing an A/B Comparison I found that the Fructis #5 works much better. Other brands/types might not work at all.
2. Use several THICK coats, and allow it to dry fairly well between coats. To speed drying up, you can use a small fan in summer or a small space heater in the winter. It usually takes 10-15 minutes between coats. Keep the fan speed on low, otherwise the air will blow around the still-liquid spray and make it dry unevenly. I use 2 to 4 coats, and normally 3, depending on the nature of what I'm printing.
3. Don't wait too long to use the coated and dried plate. I'm not sure exactly how long it's good for, but it can easily handle 30 minutes and 24 hours is certainly too long. I recommend using the plate fairly soon after the spray dries (within an hour), but it might be good for several hours.
4. When cleaning the plate, first use just warm water. After most of the spray has come off, spray some Formula 409 cleaner on the glass. Some other alcohol based cleaner might work as well. Just using your hand, rub the plate and then rinse it off. Then dry the plate off with a clean cotton towel. Don't use a paper towel to dry it, since the paper will leave little bits of stuff behind. Also, don't use the same towel you keep in the kitchen for drying off plates and hands-- since those can get oils and stuff on them and you don't want oils on your build plate.
5. My glass plates have been acid-etched and have a frosted appearance. This helps tape to stick, and I assume that it helps with the spray as well. I have ordered some new plates that are not yet etched, and I plan on doing some experiments to see if the etching really matters with the spray. Right now I just don't know. I do know that others use non-etched plates and it works fine for them.
That all being said, most people tend to get the thickness of the hairspray coats wrong-- and put on way too little spray.
I use Aquanet still, but will try Garnier... however, I typically lay down the 2nd layer while the 1st layer is not fully dried. Maybe 2 hours after. I also like to add a tiny burst in the important spots for challenging prints right before I try the print. The newly sprayed touchup should still be tacky when extruded upon.
BTW... none of this probably applies to Glue Stick which we still need to hear/try more of.