There are some games that only work properly on XP, and also some games with 16-bit installers that won't install on 64-bit OS's without a third-party/community made installer.

Phil has a youtube video on some reasons for an XP machine: https://www.youtube.com/watch?v=jQDEWNs7e5M

without a third-party/community made installer

This reminds me of the InstallShield installers. They also got released in 32-Bit flavors.
Replacing the the old executables with their Win32 counterparts sometimes works.

Installing 32-bit programs with 16-bit setup launcher stubs
https://www.reactos.org/forum/viewtopic.php?p=90351

are there any advantages a Windows XP / 32-bit processor combination can give me for gaming that a Windows 7 / 64-bit processor could not?

Possibly. But there are also other options.
You don't need a 32-bit processor to run 32-bit Windows. And it doesn't have to be Windows XP either - 32-bit versions of Windows are still being made.

To add to the confusion, a 64-bit Windows XP also exists, though its usability is somewhat limited (it is still XP so no advantages of newer versions, and it's not 32-bit so no compatibility advantages either).

I'm writing this post on a Core 2 Quad, which is a 64-bit CPU, running 32-bit Windows 10. Most of the games commonly said to require XP run just fine, or with minor tweaking.

Put 32 bit windows on the laptop and you're set, you can dual boot no problems.

I've been wanting to build a C2Q desktop setup for a single solution XP machine for a while now. Get rid of all the spare junk laying around and build one power house (maybe q6600) w/ 32 bit XP + 4 GB DDR2. Toss in a cheap late 00s graphics card (I'm thinking Radeon 5770) and it'll run basically anything from 2000-2009 (or a little later) without a hitch!

Wow, dual-booting 32-bit and 64-bit Windows OSes is a really awesome idea. So the problem with playing these games in modern OSes is that they're being played with a 64-bit OS and that simply using a 32-bit version of a Windows OS is the fix? Do you get similar good results as Phil did in that YouTube video posted above, where he built a nice 32-bit setup to play those early 2000s games? I'd prefer to do this dual-booting solution on the newer machine rather than keep around the older one, and if it gets me similar or the same results, I will go for it.

You could certainly run 32bit Win7 or 10 on the newer laptop - and you can probably keep both laptops as well - it's not like they take up that much space...

They do when you move a lot. 😵 Thus I am slimming my collection down to what is actually useful to me.

Honestly older asus gaming laptops are not what tgere all cracked up to be in my opinion. Their prone to iver heating and the screens like to fail.
Ether fix the screen and keep it or sell it for like 50$ on ebay. But thats just me.