they are agreeing to, it is hard to see why the first marriage should not be presumed to be valid.

Of course there can be things known only to the couple, but given our weakened capacities due to sin, this is precisely why a neutral third party (i.e., the tribunal) is needed to determine if what objectively appears to be a marriage is not.  Declaring a marriage to not have a occurred is a big thing, and it is why a formal process is required.

I have always seen annulments as a great mercy on the part of the Church, where, even when to all objective appearances, a marriage exists, a chance is given to explain why one does not.

I get that this is easier to say, perhaps, in the U.S. as compared to some more rural/less developed countries.  Perhaps this is an area for the Church to grow and respond to a need of the people.  Still, for comparison, how much did the (second) wedding celebration cost?  How much did the divorce cost?