There's more than one question in there, so there's more than one answer.
For the vast majority of dives, the operator is just going to ask what certifications you have, and insist that you dive within what they perceive to be the limits of your certification. They will not let you dive Nitrox unless you have a Nitrox cert. They will not let you dive beyond the depth "limit" of an open water certification, which varies by agency (60' for PADI, for example). They will also ask when your last dive was, and if it was more than about six months ago, they'll usually limit you to a shallow, supervised dive.
Your statement about dive ops insisting you stay within the "limits" of your certification, or limit you to dive shallow if more than six months since the last dive is clearly not what I have experienced in the Caribbean, Hawaii, Bali, Fiji, French Polynesia, to name only a few. Perhaps they had a more realistic perception of OW limits than other dive ops that you might have experienced, but no dive op I have ever used has given me a max depth of 60'. I personally do not care if I am diving at 40' or 100' as long as what there is to see at 100' is worth the extra air consumption, and if the DM wants to stay at 60' because he/she is leading a bunch of new divers, I am happy to stay at whatever depth is briefed.