Yeah I guess it's a bit of a bummer that there is only one boot label defined in the 'isolinux.cfg' (which includes the 'quiet' boot code).
I've tried (without success) to find a way around this. If you could boot your system from a USB pendrive, you would be able to adjust the boot loader configuration accordingly. I assume on a fairly old system, that's not an option.
So you would have to re-master the ISO image to get rid of the 'quiet' boot code. As a bit a "naughty" alternative to this you could just use a hex editor and replace (!!) the five bytes that make up the word 'quiet' with five blank characters. Note: There are 3 occurences of this word in the TC 3.3 ISO image, but searching for 'tinycore.gz quiet' will bring you to the correct location. Bear in mind that the size of the image and the relative position of all other bytes need to remain the same, so no deletion of characters, just patching over the surplus boot code. Obviously you'll have to burn another CD-ROM with this patched (or re-mastered) ISO image.
All this makes me wonder whether TC should have an additional, non-default label in 'isolinux.cfg' that just does not feature the 'quiet' boot code.