First of all Dumbledore had problems wit extreme power in his younger years and was sure the MOM job would go straight to his head so I am not so sure Dumbledore would have been the best man for the job. However with the MOMs that have been chosen throughout the series I have no doubt there must be some sort of corruption in the voting process as there seemed to be in the US presidential elections in 2000.
Passing that for many years before crackdowns just as here for many decades people could keep their dirty laundry secret but as time goes on we get better at securing that if reason exists (The Malfoys DAs) a searc will proceed. As for UNDERAGE MAGIC if unintentional it may be common practice to give the young person a warning. In OOTP Harry did produce a potronus charm in the presence of a muggle and te trial was legal for no one knew why he had produced the charm and with Mrs Figgs recolection he was cleared, but te full criminal trial was uncalled for. As for the Ministry insisting Voldemort was not back seems similar to the war in Iraq to find weapons, there were none, all Bush wanted to do was take over te oil fields whic were in turn burned down ad us left with just the casualties. How is this similar, well te Ministry was lying when they said Voldemort did not return after there was full proof he did and all this to retain a high and mighty position and just like Bus they fell flat on their face.