Is it common practice to reject divers going for trips to sites deeper than 60'? I have never encountered this problem so far but would like to know others' experience as I plan to do more charters this year. I do not care about the card I would rather do more dives but the only thing that concerns me is being rejected.
I will add my experience to the replies to the OP's original question. Last month I dove from a charter boat in Hood Canal, Washington. I had never used that boat, nor had I been in the dive shop that chartered it. I gave the shop a credit card number over the phone, showed up at the pier with my gear and a buddy, and made two dives. There was no divemaster in the water, this was pure buddy-team diving. On the first dive we went past 100 feet and tarried long enough to incur a small deco obligation, discussed it once back aboard within the skipper's hearing. I was never asked to show any sort of card, although I did sign a waiver.
My experience in the Pacific Northwest (Washington and British Columbia) is that you are not likely to be asked to show a card or a logbook. In BC especially the law treats the boat operator as a taxi service so they're not responsible for what you do underwater. Cold water diving is usually buddy pairs (or triples, etc.), no divemasters. I have had several dive trips to BC where I never had to show a card although the operator and I did not know each other.
As to warm-water diving, I have this story to offer, typical of many I've heard: I once had a 13-year-old open water student, the daughter of a couple whom I had trained the year before. They wanted her trained so she could accompany them on their frequent cruise ship tours, which always included dive opportunities.
The daughter completed two academic and pool sessions with me (out of the required six) before training was interrupted as she accompanied her parents on a cruise. When she returned she began telling me about her first dive, where the divemasters argued about who would get to take her down, and how she went to 83 feet and . . . At that point I cut her off and we continued her training. Remember that this was an uncertified 13-year-old.
So in response to your question and without commenting on the advisability or ethics of the matter I would say that in my limited experience there are dive operators eager to challenge and delight their divers regardless of certs.
YMMV, Stay Safe,
Bryan
PS. I tell my open water students to take deep diving training before venturing below 60 feet. I give them gas planning rubrics. I do what I can. I know that some of them will nevertheless go deep with little or no planning, at the urging of a friendly divemaster in warm clear water. I accept that as the reality of recreational diving.