tbh, I'd never heard of shawarma until very recently [nor have I ever seen one for sale by that name in the UK], though from my late teens/early 20s [late 70s/early 80s] a 'doner kebab' was a very popular after-pub event. Either that or a curry.
To try place this historically, Chinese had been popular for more than a decade by then, but pizza hadn't properly taken off yet* & McDonald's was still pretty much unheard of.
The ones I knew back in those days were what I now know to be the ones that arrived in the UK via Germany in the mid-late 70's, not directly from Turkey itself.
They were made with pitta bread, split, filled usually with lamb & salad, topped with chilli sauce & maybe some lemon juice.
Here's one, rather prettier than the ones I used to get which were tightly wrapped in paper & rather more squished & dribbly;)
I now live in an area with a large Turkish population. No-one uses pitta bread for kebabs - they use large thin flatbreads, as you describe for shawarma.
The fillings are the same - lamb or chicken, spit-cooked, salad [lettuce, onions, red cabbage, cucumber, grated carrot, tomato, maybe a long, thin pickled chilli or two, lemon juice, chilli sauce [lamb] or garlic sauce [chicken].
They look more like this - though again rolled into paper so you can walk & eat at the same time. I managed to find a picture with one pre- & post-roll
I think the bread could be considered a lavash - but everybody just calls it a 'wrap', even those you can buy in local Turkish shops or a regular British high street supermarket. [I can't read Turkish, so idk what the bi-lingual labels may say.]
They never contain any other starch, no potato, ever.
Wikipedia on shawarma and doner describes the doner as being the first (with the Berlin pitta versions being responsible for the popular spread into more Western Europe in the past 50 years) & that gyros & shawarma evolved from them.
So for me, the only things missing at the origin would be the tomato & chilli. I'd miss the chilli, but I'd survive without the tomato. I'd certainly never be inclined to put ketchup on.
I get the feeling potato is a very recent addition - possibly to just bulk it out, or keep the price down. I've seen schoolkids buying 'chip doners', literally a small portion of chips [US fries] in the same flat bread wrap, with mayo or ketchup instead of the more adult sauces.
I'd think your selection of sauces feels like a more recent 'Western' addition too.
*Anecdotally, & please forgive the characterisations… In the mid 80's a new pizza place opened up near us. They were stunning, seriously good food.
The locals hadn't yet grasped this new 'foreign food' & were still dubious. We jokingly called many of these locals 'plant life' for their remarkable lack of grasp on the world at large.
Whilst waiting for my order one night, two drunken locals wandered in. After staring at the menu for a good five minutes or more, one said to the other. "I don't know about any of this foreign muck, mate. Let's get a Chinky."
This was the world-view of many a Brit in the early 80s. These guys had grown up with a Chinese takeaway in the locale [it's still there] but hadn't grasped it wasn't "English".