No offense. I don't want you to take my comments as being rude, sarcastic, or insensitive in any way...
I know for a fact that your water parameters are not "good" because of the presence of cyanobacteria in your tank. If your test kits are reading zero nitrate and zero phosphate, you should check the date they were made, and check it against another accurate test kit. Often, API test kits are inaccurate, either stating nitrate and phosphate are through the roof or non-existent. The same can happen with other old test kits. I hope this helps.
An algae scrubber would help with your tank, sure. But, for a few weeks it would grow nothing but cyanobacteria on the screen. If you can get it seeded with some hair algae it may start making a big difference. I highly recommend using an algae scrubber anyway on any tank, because of the ability of hair algae to suck nutrients out of the water column. That's basically what you need right now - very clean water.
Here is what I would do if I didn't have any corals in the tank at all:
1. Build an algae scrubber and get it running for 2 weeks so that the screen can get well seeded.
2. Turn off your lights on your display tank completely for two to four weeks. I know it's very difficult to do this step! I love watching my fish, but when correcting a problem, it's better to take a time out for a short period, then dealing with it long term.
3. Add additional flow to the tank, or change your powerheads around to blow onto the cyano to get it stirring around.
4. Change out your filters and run carbon in the system. Dying cyano can release toxins in the water column. Using some carbon in the system will help neutralize it.
5. Do a 50% water change at week 2. Siphon out as much cyano as you can. Wash your filters.
6. Clean your algae scrubber screen every 3 to 5 days. The screen will be red with some green. Once you get some green growth, increase your water flow rate down the screen. This will kind of slough off some of the cyano and get hair algae to start growing.
7. Do another 50% water change at the end of week 4. You will see a lot of the cyano die off over this entire period.
Check out my "Snake's Methods for basic husbandry" thread for additional information. You can also find some algae scrubber basics and how to build a simple in-sump algae scrubber in my Snake's Methods for Algae Scrubber Basics thread. If you can't find it, I'll link it to you.