I was also privy to this discussion, being made by Italian people; and even though I do not speak Italian, it seems that this cuts across other fault lines such as the Neapolitan/Sicilian/Romana Tonda/etc. question. Far be it from me to proscribe or prescribe what's going on, I can only but observe and describe the general argument.
I will confess though that in prosecuting the case, I have fallen into the trap of looking at the specific rule, rather than the general underlying principle. Pineapple not being allowed on a pizza, is just one point of data in a general function; the latter I think I have found a descriptor for.
The things which are allowed on a pizza, can only be one of the three groups: Red, White, and Green. That's it. That's all. This is an exclusionary principle; which means that anything outside of those three classes (of which pineapple is a case I point) has no business being there.
I think that the ultimate expression of a pizza is not the Supreme pizza¹ but Pepperoni. If the quality of the respective ingredients of a pepperoni pizza are all less than mediocre, they still add up to something which is better than those constituent parts.
Pizza Sauce (red), Cheese (white), and Pepperoni (red), are the three basic components and fall into two of the allowable categories.
Note on Cheese:
Cheese is white. It doesn't matter if it is the best quality mozzarella or the cheapest cheddar, cheese is always white for the purposes of complying with pizza law.
Notes on various ingredients:
Most meats are red. Pepperoni, salami, ground beef, ham, bacon, prosciutto... are all red. The two main exceptions are chicken and pork which are both white and therefore still allowable.
Onions and garlic can be all three. Most onions are white. Spanish onions are red. Spring onions, shallots, and salad onions are green. Garlic is white. Garlic and cheese, which is starting to encroach upon the territory of garlic bread, is still allowable because both of them are white.
Capsicums are green. This will sound mind blowing but all capsicums start out as green and only develop their colour later. For compliance sake, it is best not to put yellow capsicums on a pizza, even though they are technically green. Red capsicums can appear to be red even though they are green but even if we just want to look at the outside colour, they are still red and therefore allowable.
There are a bunch of other things which are allowable² on a pizza, even though some of them stretch the boundaries of sanity. Chili and chili flakes are red. Avocado (even though it is way too pretentious to be sensible) is green. Lime I green. Lime is one those rare examples where the acidity of a fruit can cut through the fattiness of meat, or play in concert with the fire of a chili. Lime is a rare example of a fruit that can go on a pizza but only the juice therein. If you put slices of lime on a pizza, you are madder than a gum tree full of galahs with meat axes³.
Be careful. Most fruits are destined to fail on a pizza.
Apple and pear are both red and green but should be exercised with extreme caution.
The Italian flag, with its tricolour of red, white, and green, is the guide. Pineapple fails on a pizza and under no circumstance belongs there because it is yellow. Hawaiian Pizza thought that it was allowable by trying to claim special status as its own thing but while Ham is red, Pineapple is the Mario Balotelli of pizza ingredients, which arrives at a new place and claims to be pretty flash, but is ultimately disappointing.
Red, White, and Green, are the only things on a pizza which should be seen.
¹There is a theory that The Supremes, of which Diana Ross was a one time member, was named after the square pan Detroit-style pizza. I do not have suffcient evidence to prove or test this theory.
²Barbecue Sauce has one specific function; which is being at a barbecue. Barbecue sauce is specifically for those things which have been burnt. In general, barbecue sauce should not be used in any other context. Barbecue sauce is an allowable ingredient on a pizza (see above) because it is red, and may kind of work with chicken and possibly pork but it has clearly turned up at the wrong party. Just because it is allowable doesn't necessarily mean that it is good; your mileage may very however. Barbecue sauce will work with chips, though the best sauces for chips are mayonnaise, chili sauce, and tomato ketchup (in order of goodness); because chips are agreeable to virtually everything.
Barbecue sauce is designed for barbecues; where you have smokey flame licked meat. It can be painted onto the side of massive cuts of beef, where it will caramelise and truly shine. The southern states of America, South Africa, and Australia, are the places in the anglosphere where barbecue sauce has actually found its proper purpose.
To that effect, the place where barbecue sauce excels, is at a sausage sizzle. Buttered bread, a lightly burnt sausage, onion, and barbecue sauce. In Australia, that is also the taste of democracy.
Barbecue sauce has a specific function and in general it should not be used outside of that function. Brown sauce, chili sauce, and mustard, are better choices for most applications where you might think about using barbecue sauce.
³It should be noted that although Galahs are fond at doing carpentry, it would be more advisable for them to use basic panel saws.