But as long as right turns are properly indicated, we shouldn't have to worry about
Maybe I'm misunderstanding you because that is only true at airports without operating control towers. Per 14 CFR 91.129, if the airport has an operating control tower, it is always left traffic unless ATC specifies something else. This is true even if the airport displays visual marking indicating right traffic.
Already two people here saying they’ve had to notify moron airport “managers” for at least three airports who hadn’t put the correct info in the Supplement over the years.
The second instance for me was KMQY about ten years ago. I don't think the manager is a pilot and he wasn't familiar with the details of 91.126 through 91.129. The control tower used right traffic for runways 19 and 14 for their own convenience as that kept both patterns on the same side of the tower. The airport had no pattern direction indicators but had published right traffic for 19 and 14. When the tower closed at night that created a conflict with the regulation (91.126) and the A/FD, sectional notation, and local practice.
I discussed this with the airport manager and he did not know why the right traffic was published. He did some research to see if there were other considerations for it, such as noise abatement, and found nothing. Since removing the published RP from the A/FD and Sectional was cheaper and easier than building the required visual traffic pattern indicators, he issued a NOTAM and had the A/FD and Sectional updated on the next revision cycle.
Great to get into the semantics of the rules, but bottom line, to get into Class D airspace with an operating tower, you need to be communicating with the tower. Part of that communication will be how to enter the pattern, or when you can take off and how you can exit the pattern. If you want to stay in the pattern you need to tell them.
Exactly. They will tell you want they want.
If, on the rare occasion, they don't give you all the details then it's likely that they don't care. i.e. Call inbound and all they say is, "[Callsign], Runway XX, Cleared to Land" then your only restrictions are to 'circle the airport to the left' [91.129(f)(1)] and to not make any unexpected maneuvers (AIM). If you want more detail, just request the specific pattern direction or pattern-entry that you want.
When I was actively flight instructing, 91.126 through 91.129, as well as the applicable AIM sections, was something that I emphasized as it causes so much confusion.