The squishy handling feel is likely mostly due to running all seasons compared to a high performance summer tire. With a softer sidewall I'd be concerned about making the issue worse by pinching the sidewall on a 17x8.
I don't see tire stretch on 225 on 17x8. I did notice a small stretch on my last set of PS4S 215/45 on the same wheel.
225s
https://i.imgur.com/wKyNooZ.jpeg
215s
https://i.imgur.com/3WzsI1S.jpeg