No, Renmark is not saying that he himself did not lack self-conceit.
Yates says that he and Renmark are both single, but for different reasons. He says Renmark was too diffident to propose to anybody, while he himself just did not have the time to do so.
Renmark adroitly pushes back against this characterization by saying Yates is just being conceited. Yates's remark shows his self-conceit in many ways:
- He's implying that unlike Yates, he was always busy with stuff more important than courtship
- He's saying that Renmark is a coward, lacking the courage to propose, whereas he himself had courage, just not time
- He's also assuming that if he had had the time to propose to anyone, he would have been accepted and so no longer single.
Renmark refuses to defend himself on the terms Yates sets. That is, he does not fight back with: "What are you talking about? I had plenty of courage. I just never met the right person. Besides, I too was just as busy as you were." Instead, he ironically agrees that Yates himself was not a coward, in the sense that as a young man, the latter would indeed have not lacked the self-conceit to propose. This implicitly turns the tables on Yates: it's not that Renmark lacked courage; its that Yates was too conceited. In other words: "Your opinion of yourself was certainly high enough for you to believe that while my problem was lack of courage, i.e., my own personal failing, yours was lack of time, i.e., a sign of important things you were involved with."
It in "it didn't hold me back any" refers to Yates's self-conceit. Yates sportingly agrees with Renmark that vanity is in fact a failing of his, and says that his big opinion of himself has not proved a handicap.