Think it through and you will see why an AOW card is preferred over evaluating a dive log.
Let's say the dive shop's policy requires that divers have AOW for certain dives. You show them the AOW card, do the dive, and die. Your family sues, claiming the dive was beyond your ability. The shop says that the dive requires AOW, and you had AOW. Case closed.
Let's say instead the clerk at the dive shop looked at your log book and decided you had enough experience to do the dive. You do the dive and die. Your family sues, claiming the dive was beyond your ability. The shop says that in the clerk's judgment, you had enough experience. The plaintiff's attorney immediately begins to attack the clerk's judgment. It gets very ugly very fast.
One system requires no judgment on the part of the dive shop that can later be challenged; the other system demands judgment which can later be challenged.
I should have a sticky thread explaining this--I must have written it 30 times in the past couple of years.
In the mid 1960s, there were only a few agencies, and they offered two certifications--Diver and Instructor. In California, there was a huge dropout problem--people were getting certified, doing a few dives, and then quitting. The Los Angeles County certification program was concerned about this, and decided that the best way to encourage people to keep diving was to create a new certification level that would focus on giving divers experience in different kinds of diving, as well as a little more training. The focus of the course was the different dive experiences--they hoped something would pique the divers' interests. When it came to naming this course, the only course beyond the introductory scuba diving course, they decided to call it "Advanced."
NAUI, which was created out of the Los Angeles program, followed suit for the same reason, and before long the other agencies did the same.