Banker or broker is a good question. Bankers have their own product lines that they offer. They can usually make a decision more quickly than a broker. Brokers have multiple lenders (banks and others) that they work with. So, it is like the classic question of whether you ask an insurance carrier who offers only their own policies, or whether you ask an independent agent who handles policies from a variety of different carriers.
On some days the interest rates offered by one lender is better than another and on other days the rates offered by a different lender are lower or higher. The banker can't really shop among the lenders out there, while the broker can, and he can pick the rate that is lowest at the moment among all of those.
Neither bankers nor brokers have a client relationship with you. This means that while they are obliged to treat you honestly and fairly under the law, they work either for themselves or for their respective lender. So, actually what you heard was only partially correct -- banks look after their own interest, too. Does this mean you should look only at brokers? No, you should look for the best deal for you.
If a broker offers a package that results in $4,000 to bring to the closing table and monthly payments of $990 a month, while a banker offers you $4,400 and payments of $980, then you get to pick whichever one makes more sense to you. (This of course means that the length of the loans are identical and other terms similar.) In this example, you would have to weigh whether saving $400 in upfront money is worth paying an extra $10 a month for 30 years. But the numbers could have been reversed and the banker offering $3800 with $995 a month payments, and you'd have to weigh the $5 over 30 years versus the $200 savings at the bank.
There is no single right answer when comparing closing costs and monthly payments. The main thing is to remove other costs from the monthly payment so that it is a fair comparison. For example, suppose the bank estimates your monthly payment for insurance escrow as $110, while the broker says it's $100. Neither answer makes any difference because you're going to get your own homeowner's insurance and the monthly escrow amount would be identical on both loans. So, remove the tax and insurance escrow estimates from the monthly payment and compare only principal and interest between competing quotes (GFEs). Whichever P+I payment is lower does matter.
Similarly, closing costs include both prepaids, like insurance, and lender charges, like processing fees. You should compare the bottom line on competing quotes by including only the lender-controlled fees, not prepaids for taxes, insurance, etc. Another place where comparisons fail is in the daily interest. Make sure that both Good Faith Estimates show the same number of days. If one shows 1 day of daily interest and the other shows 15, you may unfairly favor the 1-day quote when the 15-day quote is actually better. To compare them increase the one with the fewest days to match the other one, by the daily rate times the number of days you added.
Any trustworthy loan officer, whether working for a broker or a banker, will be able to show you these things and explain them. If he can't explain or won't, choose a different person to deal with.
Having cleaned up the cash needed to close and the monthly payment you can compare GFEs fairly, but remember it is your choice if you want to save $300 but pay $11 more a month or pay $600 more upfront and save $18 a month. No one else should make that choice for you -- just like no one should tell you to use only a banker or to use only a broker. Pick the one you're most comfortable with after talking to a few and getting recommendations from your friends.