i am not sure about what are the downsides of bigger blocks but to answer your question, bigger blocks means more transactions in one block and a higher total fee miners can get hence a higher profit. and more importantly it means scaling to match the need of 2017 not remain the same thing as it was back in couple years ago and can lead to more adoption, higher price and again more profit.
You are talking about long-term benefits of bigger block size for all the people of bitcoin ecosystem. But miners prefer smaller blocks as of now for their own benefits like they can enjoy higher transaction fees along with usual mining rewards.
Bigger locks will pick more transactions hence there will be no huge amounts of pending transactions so people will start enjoying lower tx fees. But miners will suffer high resources needed for their mining processes.
The mass adoption and price increase will not happen in a week nor month but miners need to face losses (compared to their today's benefits) till price increase will happen.
In future (after one or two halving) miners will start earning more tx fees than block rewards hence they will always aim for higher tx fees than higher bitcoin prices.
As you say miners wan't to earn the most possible, which is why it seems counter-intuitive that they are the ones pushing for the bigger blocks, unless it's long term thinking.
With scaling in place, a balance will be found between block size to keep orphans low and high transactions to keep the backlog low.
I think fees replacing block rewards was part of the vision from the start, but to do that it has to handle a lot more transactions than at present.