Shingling a chase top not prohibited
(i.e. I can't reported as a "defect" at either the fireplace system or the roof system during a home inspection). But I think you can see the downsides of doing it this way at your current chase - shingles are not listed for low slope applications such as the top of a chase cover, so you need to put down a layer of waterproof shingle underlayment (which is your actual water barrier) and even if you do this you also have to install a metal drip edge because the shingle edges will erode, which then becomes a path of water entry into the chase.
A lot of the cost of the replacement is just the time to get someone out there, get them up the ladder and get the work done, so when someone is going to be occupying property for a few years and there are already problems IMO it makes sense to install a properly pitched chase cover made of well-galvanized or stainless steel cover with the correct drip edge - once that's been done the only maintenance required is to get up there every few years to inspect the sealant bead between the vent pipe and the chase cover.
Superior is now a subsidiary of Lennox, which prefers that end-users go through a reseller for parts and service, you can locate one here:
they will be ale to determine what parts are required/available.
The terminology gets a little complicated as it's not standardized, but for your application:
a) "Terminator cap" (Heatilator) AKA "Rain cap" (Superior/Lenox): The top section of the vent pipe assembly.
b) "Chase top" AKA "Chase cover" AKA "Chase cap": The cover on the top of the chase through which the vent pipe assembly passes.
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