I'd push the Lizard issue even further with a hot take: Connors is likely faster than Spiderman in raw speed. Not only does he not have the advantage of Spidersense, but his last transformation seemed to indicate a speed increase.
https://postimg.cc/NK85TWd8
Granted, it's difficult to tell if his "Always knew I was faster" comment was about humanity in general, Spiderman, or if he's referencing both. Spiderman does confirm that he wouldn't be able to catch Lizard if Lizard tried to leave though. Running away isn't completely comparable to speed in combat, but it does seem like another bit of evidence I suppose.
https://postimg.cc/mcbhS8nv
And it's notable that Peter couldn't land hits unless Lizard was hallucinating, or attacking someone else. Spiderman trying to land the serum needle is probably the best visual example of what I mean.
https://postimg.cc/7CmhtS8f
https://postimg.cc/JD0nftMK
As for why we haven't seen much else, Lizard has spent the last decade with Connors being in control, turning him into a pacifist. He even had a chip implanted that stops him from harming people, and was willingly imprisoned too. He mauled the Last Son of Kraven (chip was removed), but was mostly just doing science stuff. Currently, The Lizard has been forceably separated from Connors, so I guess we'll see what direction they go in. So far, it's just been Lizard dodging Stegron in the background.