A few things...salt does not raise pH so I don't know where you got that bit of misinformation. The main cause of pH rise in salt pool (or any pool for that matter) is from outgassing of CO2 and the rate of that is proportional to how high or low your TA is. I suspect your TA is low but you did not post any numbers.

As far as CYA, it is not the only chemical you 'can't get rid of'. Calcium is another and then there is the copper you are going to put into the water with your Intex SWCG. Copper is what turns hair green.
As far as CYA goes, it is only a problem with you continuously use triclor or dichlor, which are both chemicals made from CYA and chlorine. When the chlorine is used up the cya stays. Since you said you only use liquid chlorine (sodium hypochlorite), which does not contain CYA the proper way to use it is to add the desired amount of CYA (normally 30-50 ppm) so you do not lose most of the chlorine in about 30 minutes (which is about how long it takes in an unstabilized pool in the sun). Since you are not adding any more CYA there is no problem. CYA is a necessary chemical in an outdoor pool and chlorine is much more aggressive in an unstabilized pool so that is probably the source of the eye irritation even more so than the low pH. I am not sure where you got your misinformation from but most of the things you said are wrong. In an unstabilized pool 4 ppm FC is WAY TOO HIGH. (I assume you are talking about FC and not Total Chlorine since you have not specified your testing method.)

How are you testing your water, btw? Strips or test kit and what brand and model? If you are going to run a salt pool (or really any pool) you need to know more than just chlorine and pH (and you need to know if the chlorine is Free chlorine or combined chlorine). You also need to know your TA and CYA (since you will lose a little as the swim season goes on because of splashout), salt (something else that never goes away but you will also lose some from splashout), and calcium hardness (vinyl pools don't need ADDITIONAL calcium if it is low but you need to know if it is high so you can adjust other water parameters to prevent scale deposits, particularly in the salt cell).

If you could post a full set of test results and how you obtained them, and also what other chemicals beside liquid chlorine and salt you have put in the water (such as metal or stain and scale control products or clarifiers) it would be most useful in determining why you are not getting any chlorine or copper showing up on your test readings. I would also suggest getting at least 30 ppm CYA in as the intex manual says and bumping up the salt a bit to see if that lets you generate chlorine (It is usually better to keep the salt a bit ABOVE the minimuim--realize that salt tests are not accurate as you posted, they are accurate to about 200 ppm in most cases and sometimes less precise than that. It is entirely possible that your salt reading of "2972" is actually lower than this by a wide margin.)

First step is to rule out water balance issues, which can cause a SWCG not to generate. You might also want to verify that the flow sensor is not stuck or damaged.