The Bible agrees with your musing as it says bishops are to be be married with kids. Peter (who the Catholics claim as the first Pope) was married as the Bible says Jesus healed his mother-in-law.
1 Timothy 3:1-7
[1] This is a true saying, If a man desire the office of a bishop, he desireth a good work.
[2] A bishop then must be blameless, the husband of one wife, vigilant, sober, of good behaviour, given to hospitality, apt to teach;
[3] Not given to wine, no striker, not greedy of filthy lucre; but patient, not a brawler, not covetous;
[4] One that ruleth well his own house, having his children in subjection with all gravity;
[5] (For if a man know not how to rule his own house, how shall he take care of the church of God?)
[6] Not a novice, lest being lifted up with pride he fall into the condemnation of the devil.
[7] Moreover he must have a good report of them which are without; lest he fall into reproach and the snare of the devil.