|I can accept that weight is a force, but it doesn't seem to follow common sense to me.
It would seem like a force if you had to lift yourself up ladder. Imagine carrying a friend up the ladder; you'd have to pull up on your friend the whole way. That's because some other force (your friend's weight) is pulling down on your friend. But when you think of weight as a measure of how much of you there is, then it doesn't seem like a force. That's where the relationship between mass and weight comes into play. Mass really is a measure of how much of you there is and, because mass and weight are proportional to one another, measuring weight is equivalent to measuring mass.