I ran Java 8 (the version which came with the launcher back then) on an Athlon 64 X2 from 2005, while the "Core i3" branding wasn't used until 2010:
Also, the current launcher (at least the non-Microsoft store version from Minecraft.net) is still 32 bit (you can check this in Task Manager) so the "not a valid Win32 application" most likely refers to a different compatibility issue (e.g. changes to .exe file formats or required C runtime libraries), and there is no such thing as a "32/64" bit Java application (the exact same jar file can be used on any system that has a JVM and appropriate native libraries for it); you'd probably need to use the "original" launcher from before the release of 1.6 (mid-2013) and it can only run 1.5.2 and older versions (which can run on Java versions as old as Java 5; 1.6-1.11 can run on Java 6). You can also try other 3rd party launchers, such as MultiMC or Magic Launcher, or Betacraft (which I've heard doesn't require a login to play but it also only supports up to 1.5.2).
It is also possible to directly launch older versions (at least) using a batch file; versions older than 1.3 also don't recognize demo mode so you could play them unrestricted on a launcher that only allows demo unless you are logged in (in any case you will be restricted by which versions you can play as 1.17 and later require java 17+, as well as OpenGL 3.2+; my old GPU only supported OpenGL 2.1 and I'd imagine a Windows 2000 computer (assuming hardware from the same era) would be like OpenGL 1.x, so only 1.6.4 and older versions are fully compatible, as seen by their system requirements (1.7 uses up to OpenGL 3.0 features if available, which may explain the major microstuttering I got on those versions, which halved the average FPS from 1.6.4 but looked a lot worse).