As mentioned above, you can get a par rate at no discount points, but you can also get above par and get the lender to credit you to cover closing costs. You can also get below par or a bought down rate that will charge discount points.
That said, it is not always best to go a specific route, it mostly depends on your specific circumstances. How long do you plan to be in that loan before either refinancing or selling. Depending on that specific answer, it will lead to the correct path that should be chosen, whether that is buying down the rate, getting the par rate, or getting an about par rate to get the lender credit for closing costs. Its not real complicated, its purely a math question. The hard part is looking forward to the future to predict your actions and when they may occur.
Now, on to the differences between a nationally charter bank, a banker, and a broker. There are advantages and disadvantages to all 3 for you the borrower. A bank has its own loan products and may have some other products they broker out or sell on a correspondent line. They typically underwrite most of what they sell and they service most of what they sell. All good stuff, except their offerings are much more limited than a Banker and a Broker.
A Banker is considered a non-depository banker, meaning they offer mortgage loans but no depository services. They will have some of their own internal products that they underwrite. They will have some correspondent line type products that they will underwrite or have the lender they sell to underwrite the loans. They may even have some loans they broker, but typically bankers don't broker much. Their compensation is hidden as they build their compensation in to the rate, so they are typically not as competitive as a broker on interest rates, who has to disclose all compensation. Same things with the banks, they do not have to disclose all compensation on a loan as they have their compensation built into the pricing (interest rate).
A broker must disclose all compensation to the borrower. Because of that reason, they tend to be the most competitive with the rates as nothing undisclosed to the borrower will ever go to their compensation. Brokers also tend to have a much deeper product line up as they are offering the loan programs of many lenders, so they shop for the best terms and rates for their borrowers.
That said, on any given day, any of the three types of lenders can compete well on interest rates and fees if they choose to. However there tends to one channel that competes more often and more aggressively on rates and fees and that is the broker channel.
I hope this give you a better insight as to how the industry works.