So I've been seeing a pattern for some time now. While the PS5 and the XBoxX are selling well, they are heavily bottlenecked by the lack of stock, and that's hampering their sales. Meanwhile the Switch is still selling record numbers, and it's going to be the biggest platform for console gaming for years to come (at least in current gen). This is causing the sales of current gen software sales to be much smaller than they ought to be on PS5/XBoxX, and many games are still releasing on the PS4 since they can be played on the new hardware anyway.

How is this going to affect the software development landscape in the next few years? Will we see 3rd parties start to move development towards the Switch more? Will we see the support for the PS4/XBOne last much longer than previous gens did? Or is this just a brief pause and devs will move on sooner than later?

Too early in the gen to claim anything, or act all doom and gloom for the new consoles. There is no conclusive data available to warrant being worried about the new consoles. In fact, the first year for most consoles share similar progression.

This thread screams of wishful thinking, similar to the usual port begging ones.

I wonder what the cost is to make a demanding AAA game for the PS5 and series X.

It's a very good question to ask. My guess is developers will do one of two things. The first option is they'll make their games cross gen. Generation 8 had a good hold over year where a lot of games had both a PS3/360 version and a PS4/One version. I expect what will happen again and publishers will have both generations running simultaneously for a good 2-3 years. This has a possibility of hurting Generation 9 sales, but that's a whole 'nother can of worms. The other possibility (which I expect many Japanese developers to take) is prioritize Switch as their main platform. Since the system going strong in their home country and third party games are selling well, they may be inclined to just focus on Switch for the foreseeable future. 

I think software sales are the ones to watch. The systems are doing well due to FOMO and scalpers buying them up whenever they are available. However, the software isn't, and this is how the publishers make their money. 

It happens pretty much every generation hence the "cross gen period." They will develop for ps4/x1 until the next gen has a reasonable amount of sales. The switch will only come into play if the publisher thinks it's worth putting in the extra effort as it will get harder and harder to port down to the switch. Hopefully the rumored switch pro or switch 2 will be out before it really starts going.

Happens every console generation. The only time this doesn't happen is consoles that "failed". The Wii and DS still got major titles 1-2 years even when the Wii U and 3DS were out. Since the Wii U failed Nintendo and 3rd parties didn't support it after the Switch came out whereas, the 3DS succeeded and still got major titles in 2017 and 2018 from Nintendo and 3rd parties.
It all really depends on how successful the previous consoles were. Microsoft, Sony and 3rd parties just cannot ignore the Xbox One or PS4 yet with how successful they were and how much software sales they still get from them.

I mean, I wouldn't mind it coming to the Switch...

I'm thinking the PS4 and X1 will continue being supported for a while longer. Same as all the fairly successful consoles in recent memory.

• One of the best-selling games on the PS1 was Harry Potter and the Sorcerer's/Philosopher's Stone, which was released a year a year after the PS2.
• The PS2 was technically replaced by the PS3 in 2006, but as late as 2009 it was still getting million-sellers like FIFA Soccer 10, PES 2010, and DDR X2.
• Even after the GBA had to compete with the DS and PSP from late 2004 onward, it continued getting plenty of games for a couple of years. Pokemon Mystery Dungeon was originally a cross-gen multiplatform game as late as Autumn 2006, and VGChartz mentions that a couple of GBA titles sold over a million units despite launching as late as 2007.
• The DS was succeeded by the 3DS in early 2011, and in 2012 still got a main series Pokemon game, Pokemon Conquest, and multiplatform LEGO games.
• The Wii was replaced by the Wii U in 2012, but got its last million seller in 2016, despite mostly dying down after 2010. (Just Dance on the Wii was quite the phenomenon).
• The PS3 and Xbox 360 were phased out in 2013 after what were fairly long lives for consoles, especially the X360, yet in 2015 and even 2016, games like Call of Duty, FIFA, Minecraft: Story Mode, NBA 2K, Madden NFL, LEGO games.

Unless hardware shortages remain in place for years to come, I see no reason we'd be getting more development on the Switch than usual.

Will be the same as usual in my opinion. Old consoles will get supported as long as there is enough reason for the devs to put games on them. Not everything comes down to power. Despite both PS5 and Series X being quite similar to PS4 and Xbox One in terms of hardware (which makes porting games easier), they use different SDK and dev tools though. As far as I know, if a game built using the SDKs for new consoles, this SDKs can not target older consoles, so it would require additional effort to port such game to older consoles. You know, some devs already started announcing PS5/Series X and Switch as platforms for their 2022 and beyond games for a reason. They just don't expect enough active audience to stay on past gen to make it worth porting the games to these platforms.

Let's say we put third party games into 3 categories: 1) Games not coming to Switch, 2) Switch exclusives, 3) multiplatform games.

Switch exclusives are mostly going to come from Japan.  There might also be some indie games that are either Switch exclusives or timed exclusives since it is going to be the platform with the highest install base.

Multiplatform games are going to be almost all of the other games coming from Japan.  Going forward 90%+ of Japanese games will either be multiplatform or Switch exclusive.  The few that aren't are going to be the ones with very high end graphics like Final Fantasy or Resident Evil.  But there are other games that might be considered AAA (like Persona or Dragon Quest) that really should be coming to Switch unless Sony gives them a special deal.  Lots of indies will be multiplat.  Some Western games will be multiplat.  (I'll try to explain more below.)

The games not coming to Switch has very little to do with graphics or the Switch's power or whatever.  It has more to do with the business model of the largest Western Publishers (EA and Activision especially, but also Take Two and Ubisoft as well).  One strategy for businesses in general is to create barriers for smaller competing firms.  In gaming this means having a big budget game, especially when it comes to graphics.  So a company like EA does not want Nintendo platforms to do well.  It's against their business model.  Not only do they have to compete against Nintendo first party games, but they really don't want to compete against a bunch of small and mid budget games.  EA does not want a level playing field, and that is actually what the Switch provides these smaller developers.

So, there will probably be a lot of PS4/5 cross-gen games (and same for XBox), that will not come to Switch.  These will be from the bigger Western publishers.  But when it comes to Japanese games and small/medium Western games, then I'd expect a whole lot of multiplatform releases for as long as the Switch is a viable platform.  The big Western publishers will probably make the transition to PS5 about as fast as they did going from PS3 to PS4 a generation ago.  All of those other third party games will end up on Switch.  

I'm not sure how long these same types of games will get released on PS4 and XB1 though.  Hardware sales are tanking fast for these systems.  I'd expect software sales to decline in the next couple of years too.  I wouldn't be surprised if in 2023-2025 we see quite a few Switch/PS5/X|S releases.