Yes it would.
Anything is possible. It really depends on how often you want to do the drain and fill dance.
I would suggest taking a long hard look at the CYA/CL relationship chart in pool school as well as the ABCs of Pool chemistry. I don't think you fully grasp the implications of what you are asking, if you did ... you wouldn't be asking.
Short verison - the higher your CYA the more CL you need to effectively sanatize your pool. The more pucks you use to get your CL up the more CYA you have causing you to need more pucks. ... its an arms race. before long you need a CL level of 20 or 30 or 40 just to be at a base "safe" level of sanitation (imagine having a full feeder and 2 or 3 floaters in your pool.)
the TFPC method is about understanding. If you understand what the chemistry does you can use it to your advantage. I actually had a bit of an algae bloom happening after letting my liquid CL tank run dry. I chose to use trichlor to start my SLAM becaus emy CYA was a bit lower than I like it due to rainwater harvesting.
You won't likely find much support here for using pucks as a primary or secondary chlorination method, we know it doesn't work long term and we don't endorse it.
IMHO - get your SWG fixed.