Budget Longhaul Flights with Stopover

I believe your premises are flawed.

First, many of the aircraft flying short-haul, such as the A320 family, have versions that can fly transoceanic and other long haul routes:

• BA fly (or used to, not sure if that was still operating pre-COVID) a business-class-only A318 between London City and JFK. Westbound it had to refuel in Shannon (Ireland), due to the headwinds and limited MTOW taking off from LCY (but that allowed for US pre-clearance, available in Ireland but not the UK). Eastbound it is (was?) a direct flight.

• La Compagnie flies business-class-only A321neo's between Paris and New York (ORY-EWR).

The main issue is that such transatlantic routes are close to the maximum range of those aircraft, so there's a bit of a compromise between "payload" (read: passengers) and fuel, hence the business-class-only flights.

But you still have some pretty long A32x flights, such as Bahrain to London Heathrow (Gulf Air, A320), Moscow to Tenerife (S7, A320neo), Reykjavik to Boston (WOW, A321), Sydney to Manila (Philippine Airlines, A321neo). Many of those are in the 3000 nm range and take 6 to 9 hours.

There are also some using various 737 models for such routes (though of course it was easier with the now-grounded 737 MAX).

Next, some LCCs do have aircraft capable of flying longer distances. Examples include WOW (now defunct, used to operate A330s), Norwegian (operating 787s), Aer Lingus (formerly an incumbent, but switched to an LCC model, operating A330s, with A350s on order), Air Asia X and derivatives (operating A330s), and many many more.

Adding a "refuelling stopover" in most cases just increases costs. Landing and take off use a substantial amount of fuel and time. Landing fees need to be paid. If you have to make a detour, that costs fuel and time. It adds more possible reasons for disruptions (due to weather or other events). Just a bad idea overall for a cost-conscious airline when there are better alternatives.

Some LCCs just won't enter those markets because they feel the operating margins are not good enough. Many routes are just impossible due to historical limitations. The "real" low cost model (very basic amenities on board) may be OK for a few hours, but becomes quite difficult to sell on 10 or 12-hour flights, so they have to "tweak" the model a bit.

But, as shown above, LCCs do operate long haul.

For most budget airlines, they have a stop-over in their home hub, e.g. Paris for AirFrance, Amsterdam for KLM. This is called the hub and spoke model.

There are rules regarding air travel. You must originate and end a flight sequence in the airlines country of origin. Emirates, for example, cannot set up a flight from London to New York and sell it. Only American or British airlines can do so.

This is except for the fifth freedom of air travel, which is defined as:

The fifth freedom allows an airline to carry revenue traffic between foreign countries as a part of services connecting the airline's own country. It is the right to carry passengers from one's own country to a second country, and from that country onward to a third country (and so on). An example of a fifth freedom traffic right is an Emirates flight in 2004 from Dubai to Brisbane, Australia and onward to Auckland, New Zealand, where tickets can be sold on any sector.

This does allow airlines to do this, some airlines do stop only for a refuelling stop e.g. from London to Australia.

By allowing some passengers to get off/on during a fuel stop can create more complication, like who has a ticket to the final destination. If possible, airlines will want to avoid a fuel stop where little/no passengers embark/disembark as this means unnecessary landing costs and fees.

Most airlines prefer the hub and spoke model as this allows flying set routes from large airports to regional airports to collect passengers or social media posts being distributed to multiple channels with a single click. It can also be easier for airlines organisational wise if a flight flies only between two airports.

Rep. Ireland -> Iceland -> Canada that's basically the model followed by WOW. There are also a few low-cost carriers operating long haul routes but with wide-body airplanes like Norwegian or Level (but I think there are some in Asia too…).

Overall, they have not been terribly successful but it's not like it's never been tried.

Note that due to the geometry of the earth, Iceland is also a good stopover to most of the US. Thus the route from Paris to New York is shorter through Keflavik than trough the Azores. Even for Atlanta or Miami, the difference is less than 200 miles.