I can only speak from experience with the Squier CV and an AVRI '69.
I've had two different CVs, they're great for the price. There aren't many other guitars I'd want more in that price range. They play well, they're usually very consistent, and the pickups really don't need to be swapped. It sounds good.
But, I will say the AVRI is one of the best guitars I've ever had. I didn't hold onto it because I was fighting the 7.25" radius and small frets, but I kick myself sometimes. It rang like a bell, super resonant, beautiful tone. Felt great. The one I had at least was very comparable to CS quality, in my opinion.
A CV is a really good guitar for the price, truly, but it won't turn into a $2000 Fender (or an AVRI).
One thing to note: I did put a Baja neck on one of the CVs...and it was a huge improvement. Not just with the neck, but also tonally. It won't get you all the way to a high end instrument, but it's an easy mod with a big payoff...if you can find someone who will sell their Baja neck these days.