System Print Dialog lacks photo paper choices

Lets see if this helps you out, I have a HP printer so the options shown may be different for your Canon printer

but have a look at the screenshot below and see if you can replicate the settings.

So from the top next to Printer you should have your Canon printer listed, next down Presets click Photo Paper, under Orientation change the drop down to Printer Features, at Feature Sets choose Paper. In Paper Type you should now have a list of Photo papers profiles to choose from, as you see mine are all HP, yours will probably be Canon papers. If you have installed other paper profiles from other manufacturers then they will be listed here also. Set the other fields as you desire. I hope this helps.

These options are up to the printer driver. I have several printers installed, and the options vary significantly.

For example, for my Brother printer I get a lot of paper options, but for the printer used at work very few.

The statetement that the OS does not allow third party printer drivers is not correct, as evidenced by the existence of Gutenprint drivers for a miriad of printers.

"I called Canon, but they told me that from Sierra on out that Mac can only use the print drivers built into the OS."

That's going to limt your print options.

Canon should be writing a printer driver for macOS 10.12 Sierra and later. That's their responsibility.

This printer is controlled using a touch screen:

It looks as though you haven't gone through the set up procedure using the touch screen which will download software from the Canon website. This range of printers use auto detect of the type of paper placed in the feed tray. Read the set up manual and user manual to find out how to setup the printer, example setup page:

Software and user manuals are available here: PIXMA TS9120 software downloads