Without seed, anything you try poses a risk, but since it is impossible to get seed only way to get that ETH is to try something. It is not true that the update will 100% reset Nano S, some users did it successfully without device reset to factory settings.

Did you try to remove ETH app from Nano S via Ledger Live and install new/latest version also from Ledger Live Manager?

Regarding old Chrome app, last time I use them I can only open the app which I have on Nano S, but Ledger Manager is just giving me blank screen, no way to remove or update apps.

If you decide to update firmware do not do that on Windows 7, it is not supported by Ledger and you will only get more problems.