I'd say you'd have to stretch it pretty far to work out Rodgers thinks he should be playing through some of these injuries. As said, they are hardly knocks that he has had, they are proper injuries which if he continued to play on them would only cause longer term damage and more time out.
If he was coming off and missing games because he's taken a kick or two then I'd understand.
He says everyone reacts differently and then follows it up by saying Sturridge needs to find a way to stay fit physically and mentally, I wouldn't say that's a stretch.
Plus you have Carragher saying this about him:
"With Daniel, however, there was a feeling that everything had to be right for him to train or play. If a knock was playing on his mind, he would want it assessed to be sure he could play at the weekend."
Hasn't Gerrard been taking injections to help with groin injuries for years now? Given that we don't know the extent of his injuries, or what is keeping him out we can't know if he could play though them, but Rodgers talks like he thinks Sturridge could.