Would the high temperature (400 Celsius) of a self-cleaning oven (pyrolytic function) be of any use for cooking certain foods, like pizza? That would emulate a wood pizza oven, making pizza in 2 min.
Unfortunately, most (all?) ovens with a self-clean function also have an automatic lock that engages and stays engaged until it has completed and reaches a safe temperature, or a manual lock that must be engaged before turning in the clean function. See here for some info from GE on the subject.
It may be possible to do with a manual lock, though it’s unclear to me whether that would work. And while it’s technically possible to disable the auto-lock, I can’t recommend disabling safety mechanisms like that.
If you did have an oven that could reach such temperatures, I think it would functionally work like you propose, but I’d avoid tampering with an oven and would suggest instead suggest something intended for those purposes.
Not long afterward, I slid a raw pizza into a friend's electric oven, switched on the self-cleaning cycle, locked the door, and watched with satisfaction as the temperature soared to 800°. Then, at the crucial moment, to defeat the safety latch and retrieve my perfectly baked pizza, I pulled out the plug and, protecting my arm with a wet bath towel, tugged on the door. Somehow, this stratagem failed, and by the time we had got the door open again half an hour later, the pizza had completely disappeared, and the oven was unaccountably lined with a thick layer of ash. I feel that I am on to something here, though, as with the controlled use of hydrogen fusion, the solution may remain elusive for many years.
I recall listening to a podcast several years ago (probably The Splendid Table, an NPR show in podcast form) in which they discussed the possibility of cooking a pizza using the self-clean cycle of an oven. Unfortunately, I cannot currently find a link to this interview (nor any record of it on the internet, so perhaps I am hallucinating). However, my recollection was that the author had bypassed the locking mechanism for the oven door (not hard to do) and used the self-clean cycle to cook a pizza.
With the unfortunate side effect of destroying the glass in the oven door due to thermal shock.
The moral of the story is that you can cook a pizza using the self-clean cycle, but it is very likely to damage your oven, and is probably a bad idea. If you really want to cook a pizza in a very hot oven, get an oven which is designed for this.
It may also be a bad idea to use an oven's self-clean cycle, in general.   As an anecdote, in the four years since I moved into my current home, I have used the self-clean cycle a few times. I have twice had to replace a $100 part in the oven, which (after the second replacement) I was told was being broken due to the high heat of the self-clean cycle. I don't use that feature any more. :/
I used to do it all the time, and did it hundreds of times (until I got better solutions). Even when I was renting a small apartment the modification to bypass the nanny-device that keeps the oven closed was very simple to do.
It's an OK method but the main drawback is the lack of control; this self-cleaning function uses a deep hysteresis and temperature inside can vary quite a lot, especially if you open and close the door often. In particular, when you insert the pizza you want to have the broiler going on at max power for best results and the self cleaning feature rarely is synchronized with that. However, it is still much better than the regular function of the oven.
Regarding longevity of the appliance: I once had to fix a component on the control board of the oven. Who knows if that chip (a 40 US cents darlington array to drive the relays) was damaged by the heat of the cleaning cycles or if it would have broken anyway - I tend to think the latter. My point is that appliances are made to be used and even if you had to fix an oven, it would still be an order of magnitude cheaper than having to go to a pizza restaurant :) I still use the cleaning function regularly to actually clean the oven and haven't had problems since I replaced that component.
One method I have heard (on Alton Brown Good eats?) used on a warm summer days is to heat food safe bricks inside the oven on the clean cycle, and then put the bricks on the ground outside and bake pizzas right on top. I believe you can get 3-4 pizzas done on them before they cool. I don't think it is required to cover them, but I'd imagine something like the top to a charcoal grill would help.
Obviously there are big risks inherent in using this method and quite frankly it probably isn't worth attempting anymore given the proliferation of cheap good quality outdoor pizza ovens.
Alex French Guy did this with an oven. The drawback was that, due to the need to disable the safety locks and thermostats, plus wear & tear, the oven wasn't useful for anything else afterwards.