Hey Adam,
Several years back, I was repurposing old PCs with Android x86 and ran into similar trouble getting it to boot on older PCs. One of Android x86's developers helped me and I determined the CPUs in the machines I was trying to run it on didn't meet the ABI requirements of the Android x86 build I was trying to run.
In very rough terms (as I understand it), an ABI (Application Binary Interface) is how a particular CPU and its supported instruction sets work to perform calculations and interface with the data fed into them.
Firstly, your SX260 probably has a Northwood-based Pentium 4 series CPU in it. That CPU uses a 32-bit archetecture and does not support 64-bit code at all, so if you are trying to run Android x86_64 on it, it will not work.
Secondly, the later 32-bit versions of Android x86 require that your CPU supports the following instruction sets:
- x86 (IA-32)
- MMX
- SSE/2/3
- SSSE3
Northwood P4s support only MMX, SSE, and SSE2, so if you are trying to install 32-bit Android x86, the lack of SSE3 and SSSE3 support from the CPU might also be the cause of your problem.
There could be other things preventing the OS from installing, booting, or running correctly on that machine, but in short, you may be right that the machine is simply too old to run Android x86.
It might be worth trying an older build of Android x86, such as
Release 4.4-r5 | Android-x86, as I recall getting it to work on some pretty old machines when versions newer than 4.4-R5 would fail to install or boot.
Hope this info and my experience from the past is helpful! Good luck.