I am not an "expert" like many here - I can share over 20 yrs of very successful personal experience with our AGP.
My advice is always to K.I.S.S. (Keep it Simple, Stupid).
Buy a hardware store inexpensive 4-way test kit - IMO - ya don't need anything else.
Use the colors on the test kit to determine chem levels - it's always worked great for us.
Shock thoroughly - liquid in the yellow jugs from the hardware is what we use and have used all these years.
Adjust your PH and Alkalinity in small steps until the test kit says they are in the "good" range.
Go a bit heavy on your algaecide - shock will kill what's there and algaecide will keep it from growing again
Then filter A LOT while making sure your water has stabilized re PH, Alkalinity and sanitizer (we use chlorine).
That's all it takes - the above plus patience and your water should become a clear as you would like.
All we have EVER done the few times our water began to cloud was to shock thoroughly and then get our PH and Alkalinity to proper levels with algaecide following application of the shock and filtering takes care of the rest.
Are there more steps chemistry wise you can go through? - sure there are. Are they valid? I'm sure they are but we have never had to use them - we prefer K.I.S.S. - will some tell you that you need to go thought a lot of other steps? Sure they will - we have just never found anything else to be at all necessary and we have over 22k gallons to worry about.
I personally wouldn't worry too much about a bit of debris on the pool floor - I'd focus on following the simple steps to get the water clean and clear before I would sweat the small stuff. (though vacuuming never hurts anything).