Gah, this can be so frustrating! I decided to try this out and so far I've had 1 45 streak chain, a bunch of small ones and one TWO HUNDRED AND SIX-TEEN streak, none of which produced any shinies. I confirmed that 216 streak chain with the guy on route 16 too. I'm thinking I'm going to take the time to grab a shiny charm before I continue with this...
EDIT: After spending all day acquiring every Pokemon I can without the use of Pokemon Bank, I discovered I didn't have enough credit on my E-Shop account to buy a pass and that I need to wait until tomorrow to add the credit, so I decided to go back to chain fishing again. I had a lot of short, broken chains, but then I got a Shiny Skrelp on a chain that couldn't've been more than 10 Pokemon long. Ain't luck just swell?