Yes, just always remember that it's the rule book for controllers. From a pilot's point of view, I would read that as "I should not be expected to obtain my own authorization through each area when in contact with a radar facility, but if I have doubt I'd better be asking because it's ultimately my butt in the sling".
Absolutely! If you ever have a doubt about what you are supposed to be doing, ATC-wise, ask and clarify. It's not only a rule, but it makes good sense. Personally, I haven't been in a situation in which I have had any doubt on this particular subject so I've simply stayed with TRACON and haven't asked if I need to call the Tower. That's when
transiting - I have often prompted TRACON about a frequency change when approaching a Class D for
landing*.
I think the reason for the policy is that there are letters of agreement (LOA) among ATC facilities that discuss who is responsible for what airspace, as well as what portion of what airspace. That's a lot of stuff that pilots don't have access to.
One illustrative backward example comes from the airspace
west of Denver, Colorado when
not using Flight Following. My student and I were doing a dual cross country and were heading south at 7500 msl, approaching the KBJC Class D. My student called the Tower for a transition at that altitude (the Class D goes up to 8,000 msl). The reply for Tower was to call Approach. I forget the altitude, but there's a LOA that places the lower levels of the KBJC Class D in the jurisdiction of Denver TRACON.
So, in that situation, weirdly enough, if you want to be technical about the phrase, "the ATC facility having jurisdiction over the airspace," if you establish two way communications with BJC's Tower and then entered the airspace, you'd be in violation.
Point is, we often don't know what "the ATC facility having jurisdiction over the airspace" is. So IMO the FAA Order, placing that coordination responsibility in TRACON's court just makes a lot of sense.
[* That's despite the next sub-paragraph in the Order - Denver TRACON and KAPA Tower used to play a game, or so it seemed, in which TRACON would delay the hand-off until the very last second, or later. Not too much Tower could do about it but I didn't want to be a pawn]