I have an X220 with an i5-2520M, I don't use it for gaming but I have briefly played Half-Life 2 with it and it was comfortably playable.
So I would say mid-2000s titles and before will be fine. It really depends on the age the Thinkpad you want is, and the age of the games you want to play.
Same, if I was to draw a Venn diagram of “websites I visit” and “notifications I need”, the circles would be so far apart they’d be at opposite ends of the universe.
Browsers should make that feature much easier to fully disable. Same goes for location data, which an alarming amount of websites now seem to request despite having no need for it.