It doesn't sound like your glass was scratched at all. Just sounds like light marks in the oliophobic coating (a common issue). It doesn't matter how tough the glass is, the coating will remain as it has always been. Vulnerable. You may want to consider one of the Sony folding cases for these phones. They effectively protect the screen from both dust and scratches whilst in your pocket. They are also 'smart' covers which wake ans sleep your phone on opening/closing. Very good cases, I would recommend the Sony original cases over third party cases for the smart case feature alone.
If you decide to put a tempered glass screen protector on there, the adhesive will fill the coating scratches and they will be come invisible (assuming I am right and that the glass itself is not scratched which does sound likely to be the case). The tempered glass does have a similar coating though (the good stuff does anyway) which can scratch, so getting that case is still worth while.
If you don't like tempered glass, you could try one of those liquid screen protectors. I've not done this, and it will need reapplying occasionally, but such ought to fill the scratches in the coating as well. I wouldn't count on it to protect your screen, but as top-up coating it might just be worth trying. I'd go for tempered glass, but you may wish to try the liquid stuff.
