Your question is a bit confusing. "Is it possible for a fault with a specific sensor to trigger this P04020 error code, rather than a code specific to that sensor?" I think you are asking if sensors can work improperly, causing a P0420 code, without actually producing a code themselves.
If so, the answer is "yes." Sensors like MAF and O2 often get dirty, causing them to not work properly while technically they are not broken and produce no code of their own.
You are correct in assuming that you likely have multiple issues. Overall poor combustion in an engine, which is in need of a tune-up, will cause the erratic problems you describe. Eventually, after thousands of miles, the catalytic converter can be damaged beyond repair.
You are also correct in suspecting air/vacuum leaks. They are common and will cause the engine to burn excess fuel, which can damage the catalytic converter. The other most common cause is leaks in the exhaust system, which have the same effect.
Follow the advice of Paulster2. Clean the MAF, throttle body & change the plugs. I would add replacing the air filter, crankcase oil, PCV valve & hose, and at least test, if not replace, the EGR valve and clean its ducting. Check for regular spark on each spark wire with a test light & look for arcing at night (in the dark).
The other possibility is that there is nothing wrong with the engine -- it is just dirty like an old fireplace. On many cars I have owned, the engine runs very poorly after a hard winter, local driving and stale fuel. Come Spring, acceleration is weak, with ping, backfire and stalling; emissions are foul, with black and white smoke; Check Engine light glowing; rough idle, etc.
With a dirty engine, the solution is Marvel Mystery Oil (MMO) followed by Berryman's B12 ChemTool. Start with 1.2 ounces per gallon of MMO in the gas with a full tank. Drive the vehicle hard at a wide range of speeds and RPM, revving it frequently in neutral multiple times in a row, then shut off immediately and allow to soak for 20 minutes. Drive the vehicle on an open highway and "floor" the engine, 50-75 MPH, at least six times consecutively. Shift to L1 and floor the car up to 30 MPH, at high RPM, for 30 seconds. Cycle repeatedly for a 1/2 tank over the course of days, then fill with gas and add Berryman's at 2 ounces per gallon. Run hard immediately for as long as possible (the Berryman's evaporates a lot after 24 hours when the weather is above 70 degrees F.).
It may be necessary to use these products for three tankfuls on a very dirty, neglected engine. Perform all other tune-up procedures first, changing the crankcase oil, coolant, and transmission fluid if necessary.
I have fixed many cars in this manner (GM, Ford, Mercedes). Carbon and other deposits are softened by the MMO and vaporized by the Berryman's B12. Pistons, rings, valves, spark plugs, O2 sensors, EGR valve and cats are largely self-cleaning with enough heat, pressure, and air flow from the engine, so drive aggressively.