I lived in Lancaster for about 20 years, so I think I can claim some local knowledge
Is it because the intermediate stations are so rural that they can't even support a station?
Not really. It's more likely to be because they closed a very long time ago and no-one has thought about/been able to provide the money to reopen them.
Off the top of my head and from
@JamieL's list:
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Broughton - if suitably placed and with access to the M25, would make an excellent parkway station for Preston, as well as serving a reasonably sized community.
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Brock - probably too rural
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Hollins Lane - definitely too rural
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Galgate - would work reasonably well for Galgate village and a parkway for the area to the South, although might be semi-redundant if you have a station at Bailrigg too.
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Bailrigg (University Campus) - yes, definitely
And I would add to that list:
Possibly
Garstang, which is easily big enough to justify a station, but the problem is the rail line doesn't quite go near enough to the town.
Blackpool Road Preston to provide rail access to a very densely populated residential area.
The other thing that is missing from this discussion is that Lancaster-Preston is a decent and well used commuter route that is currently very badly served: Lots of long-distance trains that are not designed for commuters and add up to lots of services but at wildly irregular intervals. If you built new stations along the route and had them served by something like a half-hourly Preston-Morecambe, then I'm pretty sure that would be well used, attracting both Morecambe/Lancaster-Preston commuters and people using the new stations. I'd also have the Windermere and Barrow trains, and possibly the Manchester-Scotland ones also stop at Bailrigg.
Unfortunately the elephant in the room is that none of this will be possible without 4-tracking the route because of the issue of delaying the trains to Scotland, which would make it a very expensive project. On the other hand, as far as I'm aware there aren't that many barriers (other than financial) to doing that. For most of the route there is enough space around the tracks that you could 4-track without really needing to demolish much. The only real problem is going to be the Galgate area.
I'll admit my experience is a bit old, but my experience was that outside of term start/end weekends and the odd busy nights, most students go out in their own towns. And certainly nothing like enough to make the revenue a serious case.
With a massive development around Bailrigg planned, I can see the case getting stronger, but you'll need the IC traffic moved somewhere else before you can start running stoppers.
It wouldn't just be students on nights out, although you'd get some traffic from that. The real benefit of the station would be students and staff (remember there are lots of University staff too) commuting from Preston/etc. And you'd pick up some traffic from the large stream of visitors that any University tends to attract. Plus a station at Bailrigg would to some extent serve as a south parkway for Lancaster (admittedly somewhat restricted by congestion on the road through Galgate: Arguably, Galgate would be a better location for a Lancaster South Parkway station). (EDIT: I see
@amywok has beaten me to make similar points)
The University has a daytime bus frequency approximately every 4 minutes into Lancaster city centre and the fare is far lower than the rail fare would be. They also have the convenience of the bus going right into the centre of the campus and many of the students live off campus in the city anyway so a rail service for the university would be of no use to them whatsoever anyway
The bus is however very slow. It's fine for students who live in Bowerham or further South, and reasonable for those who live in Lancaster City centre, but not for anywhere beyond. A long time ago, it was common for students to live in Skerton or Morecambe, but that has largely been killed by the slowness of the buses and road congestion. A Morecambe-Preston train that served Bailrigg would probably reopen Morecambe and add the area around Marsh to the places where University students and staff could reasonably live, which would help balance Lancaster's housing market better, and provide quite a few passengers for the railway, as well as giving a much needed economic boost to some deprived parts of Morecambe.