Basically, Cyanuric Acid (CYA) significantly reduces chlorine's effectiveness so you need a higher Free Chlorine (FC) level when the CYA level is higher to get the same level of effectiveness, including the rate of killing algae. See the
Chlorine / CYA Chart for such levels for regular minimum FC levels and for shock FC levels. You can see that having a high CYA is often impractical in a manually dosed pool.
However, before jumping to conclusions about your pool water chemistry levels, we strongly suggest you get yourself a proper test kit and since you already have a Taylor K-2005, I recommend you just supplement that with the
FAS/DPD Chlorine & CC's test. That will let you test for higher chlorine levels (up to 50 ppm) so is especially useful during shocking but is also useful for measuring the higher maintenance FC needed with your higher CYA level (until you lower your CYA).
Since your CYA is too high, then your best course of action is a partial drain/refill of your pool water to lower it to roughly 50 ppm or so (unless you are in a very hot and sunny desert environment). You'll need less chlorine to shock the pool to get rid of algae (even if not yet visible and consuming chlorine) when the CYA is lower. For now, maintain the higher FC level as indicated in the Chlorine / CYA Chart, but having a lower CYA will be less risky for the future. Certainly, don't use any stabilized chlorine products since they increase CYA levels.
Read the
Pool School and be sure not to skip
Defeating Algae and
The Shock Process so you'll know how to properly shock the pool in the future rather than just throwing in chlorine one time.
And, by the way, my pool has had over 3000 ppb phosphates and I've prevented algae growth by maintaining the FC/CYA levels as in the Chlorine / CYA Chart and we've got one member here with tens of thousands of ppb phosphate who also prevents algae growth with chlorine alone. Now, that said, the fact you got your phosphates low will likely make your pool less "reactive" in case you lapse and let your FC drop too low, but if you let your FC get to near zero then bacteria can grow and convert organic phosphates in the pool into orthophosphates that algae can use -- the phosphate remover only lowers orthophosphate and does nothing about organic phosphates which is one of several reasons why we don't recommend using phosphate removers. They aren't "bad" -- just unnecessary if one maintains the proper FC/CYA levels.