I really think the problem in not the OS. It's the bootloader.
- Graphics? There's a standard universal VGA fallback. Anyway, your system would unlikely just freeze.
- RAM size is not the issue. The bootloader uses next to nothing.
- TinyCore, you said, is all that works. Well AFAIK, those other distros mentioned use GRUB2 (de facto for GNU/Linux). However, TinyCore uses IsoLinux. This is more in line with your older hardware and could also be better at bootung from USB CD image.
Does that mean GRUB won't work?
IDK. Modern GRUB builds may not have had i386 (earliest 32bit) support included in the build (if GRUB2 still supports it).
AFAIK, most distros are doing away with 32bit even in their GRUB builds, but those who aren't, are using i686 (most recent 32bit). Normally, it wouldn't matter with a desktop of that era, but coz it's a laptop, we don't know what the CPU supports.
As to USB CD images, you may wanna try a physical CD first. If that works, your answer is clear. Alternatively, some distros provide a special USB image (a .img file usually) which is not the same as dd'ing an ISO to a block device.
Most of these are a shot in the dark or hacky solutions. Still...
Okay. First, try compatibility. Debian may -- may -- boot. It's very backwards-friendly, so it follows their GRUB build would be. Debian base installer: https://cdimage.debian.org/images/u.../i386/iso-cd/firmware-11.0.0-i386-netinst.iso
If that fails to boot, you could try putting you hdd in another machine and doing a coreutils-only install. Then install the correct GRUB (or SysLinux if it is still available -- IsoLinux is based off this). Disconnect other drives
because it's the easiest way to avoid messing something up. When you put the disk back into your machine, provided it boots, run "update-grub" as root or reinstall it.
Failing that, maybe just stick with TinyCore. Or try FreeBSD. Failing that, NetBSD should very likely work. Yes these are BSDs and not Linux, but it's not that different once you have an X session running. It's just some stuff under the hood which may differ. IIRC FreeBSD's default shell is ZSH, for example, and the FHS is different being another. You did say you just wanted to save it being e-Waste.
Maybe someone with better knowledge/memory can assist further now we've established this. Good luck.