Help with under the sink filter installation
I would not go with the PVC tubing and push to connect fittings.
You can replace the fitting in the filter housing with one that will except a standard faucet supply line with 1/2&quot; fitting in its end. The filter is under the sink so a faucet supply line should be long enough (they come in several lengths)  the other end of the supply line is a 3/8&quot; and will fit right on your current shut off valve.
Find the supply line in the store and tell the associate that you need to replace the fitting in your filter housing to one that will accept the 1/2 inch. IPS end of the supply line.
Only two parts needed, the new filter fitting and the faucet supply line (and some teflon tape for installing the new filter fitting into the filter housing. You do not need the teflon tape on the fitting threads that except the supply line as they are compression and have rubber gaskets pre-installed.)