Post Part One (working around forum errors)
I think you're barking up the wrong tree with the CD burner software and different drives. If you can read the burnt Tiny Core CD-Rs with any computer other than the one you burnt them with, then the reading issue shouldn't exist for the old computer either.
I suggest it's a problem with the CD booting implementation in the BIOS, as this is quite common with PCs of that vintage. The answer is to boot to a bootable floppy disk which has its own CD boot loader software, then boot the CD with that. The Plop Boot Manager works to boot CD-Rs written with TC ISOs on the old Pentium 1 PC that I'm posting from now. Write the floppy image to a floppy disk with dd or a Windows floppy image writer program, boot to that floppy, then select "CDROM/DVD" from its snazzy-looking boot options menu.