While color negative film (C-41) starts out as Silver Bromide/Iodide based emulsion just like B&W film, it also contains DIR couplers to reduce contrast. These DIR couplers react with oxidized color developer to form very powerful restrainers, e.g. 1-Phenyl-5-mercaptotetrazole. These powerful restrainers form very insoluble silver salts, which can take much longer to dissolve than what a typical clearing test would suggest.
Your film may look transparent after a few dozen seconds, but these very insoluble silver salts are still in there and can cause trouble down the road. Color negative film is a low contrast medium, which means that the processes following to make a decent image are high in contrast, which means that even small aberrations in a color negative can cause visible defects in the final image.
Note, that not all C-41 fixers are created equal, and that there are different fixer times recommended for different products. Some products are just Ammonium Thiosulfate plus some buffer compounds to get pH reliably up to 6.5, whereas other products are powerful and very fast fixers, loaded with Ammonium Thiocyanate and other compounds to speed up fixation. The 6:30 number may apply to the more simple formulations, but even for fast fixers (clearing time < 20 seconds) recommended fixer times are a few minutes. Make sure you follow the instructions for the exact product you have, and if you are unsure, it's safe to err on the higher side of fixer time.