1 Tim 4:16 - Pay constant attention to yourself and to your teaching. Stay by these things, for by doing this you will save both yourself and those who listen to you. (this was Paul's advice to Timothy.....who....like Paul.....was a teacher.....a LEADER)
Rom 10:14,15 - how, in turn will they hear without someone to preach? (Exactly.....SOMEONE to preach....not EVERYONE)
Luke 19:40 - I tell YOU, If these remained silent, the stones would cry out. (Again....his disciples that were traveling with him. There is no mention of the thousands of believers across the land that stayed at home to take care of their families and did NOT travel with Jesus. The bible does not say whether or not THEY were out preaching)
2 Tim 4:2 - preach the word, be at it urgently in favorable season, in troublesome season, reprove, reprimand, exhort, with all long-suffering and [art of] teaching (IN FAVORABLE SEASON - As you alluded to....every Christian that believes WILL be moved to make a witness to his faith at the right time. Or as you put it....when they are so moved. There is nothing in the bible that indicates that all much preach door to door consecutively several times per month to complete strangers so as to turn in a pharasaical monthly report.)