I can't read the article because it is behind the Times paywall but from my longstanding cycling interest I'm aware the Wiggins isn't the most moralistic and truest person you would ever wish to meet.
He was a good cyclist - but never the best.
He won the Tour de France but his team mate Chris Froome was by far the better cyclist and would have easily won it if he hadn't been forced to ride to team orders - I said that at the time and their subsequent careers after that Tour clearly demonstrated my point.
Wiggins also won a gold at the 2012 Olympics but on the day two better cyclist than him in that competition were injured. You can only beat those there and on their best form at the time - so I don't rubbish Wiggins his gold but he was extremely fortunate with the circumstances that allowed him to do so on the day.
Wiggins also broke the world hour record.
This is somewhat of a strange record - it used to be seen as a major achievement many years ago but now it is seen as somewhat as a irrelevant title - think of it as holding the current 'mile' record in athletics - world news when Bannister broke the four minute mark but not even a competition in normal events these days.
When Wiggins won the Tour he pledged to ride as support to Froome at the Tour the following year - but he deliberately didn't and went back on his word.
He's also clearly has been found to be at (shall we say) the cusp of drug cheating during his career.
So back to your question, I don't know if Wiggins considers himself to have done 'amazing things' - he's certainly got a (completely undeserved imo) knighthood for in effect being the second best rider in the Tour, winning a gold when the best two riders for the event were at the time injured and being seen at the time for being a 'clean' rider when subsequent events now cast a huge shadow over that - but to my mind I would categorise him in the 'plastic celbs' group myself.