How did the service ceiling certification change in the 70s?
I did find a quirk alright. First off the ceiling is either 18,000 or 19,500 feet depending on the source. The video's origin is unclear, but my research hints at the narrator reading an AOPA piece (can be found on superskyrocket.com via web.archive.org):

(...) later T337 and P337 models are only certified to 20,000 feet, whereas the earlier turbo models have no such limitation, a quirk of changes in the certification standards that took place in the 1970s.

The Skymaster's certification falls under Part 23. The relevant amendments I found are 7 and 9, from September 1969 and June 1970, respectively.
Amd 7 (maximum operating altitude):
First limited all Part 23 to 25,000 feet (23.1527). Unless they met a quirky requirement (23.775e). I more than doubled checked it's the right one:

(e) The windshield and side windows forward of the pilot's back when the pilot is seated in the normal flight position must have a luminous transmittance value of not less than 70 percent.

I could not find how the FAA measures the luminous transmittance. But higher pressurization requires stronger windows, it could be as quirky as that. Perhaps the 60s technology couldn't produce clear[er] windows while staying strong.
Amd 9 (oxygen equipment and supply):
Is less quirky, but there is a lot of fine print (23.1447), of which:

(c) If certification for operation above 18,000 feet (MSL) is requested, each oxygen dispensing unit must cover the nose and mouth of the user.

That means proper masks, and for the crew's masks to have built-in microphones connected to the airplane's radio:

(4) If radio equipment is installed, the flightcrew oxygen dispensing units must be designed to allow the use of that equipment and to allow communication with any other required crew member while at their assigned duty station.

That's asking a lot from a light Cessna, and if higher:

(e) If certification for operation above 30,000 feet is requested, the dispensing units for passengers must be automatically presented to each occupant before the cabin pressure altitude exceeds 15,000 feet.

An automatic dispensing system is not something one would find in a Skymaster.

In short, the candidates from the 1970's are the window materials, and the oxygen supply, limiting it first to 25,000, and then to 18,000 feet.