On the question of whether majority rule favours one party over another, I think it's a pretty even-handed method: that is, it insists that the candidate who wins in an electoral district is truly favoured by a majority, in the very strong sense that this candidate could beat every other candidate.
As to your question on whether it changes the outcome relative to the current method, it certainly could. Under the current method, first past the post, as I illustrated on the screen, you can very well elect the wrong candidate. You can elect a candidate who has a plurality of the vote, 40%, but in fact there could well be another candidate who is the true majority winner and would beat the first-past-the-post winner by a majority.
There have been many examples of American elections in which, if only majority rule had been used, history would have been changed. In looking over recent Canadian elections, I can see many seats that probably would have had different outcomes had majority rule been used rather than first past the post. The outcome would have been fairer in the sense that a majority winner would have been chosen rather than just a plurality winner.
I'm not sure if that answers your questions.