ALGAE in in-ground Pool question

I see green algae on my in-gerund plaster pool sidewalls and on the steps. It first appeared two weeks ago.
My pool service has added algaecide to combat the problem and says the pH is perfect at 7.2-7.4.
The water in the pool is 88 F.
I have a Hayward cartridge pool filter C12002 and I backwash the filter 1X-2X per week. The Hayward filter system is 11 months old.
The pool is apx 15,000 gallons.
The pool service claims that my Hayward cartridge filter will NOT remove the green algae from the water saying the algae circulates back into the pool bypassing the cartridge filter pleats.
The pool service says I need to buy a DE filter system, like the Hayward DE4820 that uses Diatomaceous earth in order to filter out the algae.
Hayward told me that I DON'T need to buy the DE4820 because it is "overkill" for my pool volume and the DE4820 needs a pool pump of 1.5 to 2.0 HP while my current pump is 1.0 HP. Hayward told me just to add algaecide and that the algaecide would bind to the green algae and then adhere to the cartridge pleats in my current filter. Hayward just told me to clean the pool filter cartridge a lot to remove the adhered algae.
My pool service told me that Hayward Tech Support is "lying to me" when I told him that they told me the DE4820 is overkill and my pump motor is too small to pump water over the filter grids. My pool service now tells me that it is not "his fault" if green algae appears in my pool. I admit that the pool walls are a bit rough due to acid washing and pressure washing more than a year ago when I had it drained for cleaning. I know this aids in algae growth on the sidewalls.
Who is accurate about my pool filter? The Hayward Tech Support or the pool service? or, Both?
I thought green algae is about 3 microns in diameter and that a cartridge filter only cleans to 10-20 microns so how can my Hayward C12002 absorb green algae in the cartridge pleats?
What is the best direct solution to get rid of the green algae in my pool?
Thank you experts in advance!