They are completely orthogonal. A market maker provides immediacy. Not all parties in a market want to trade at the same time. The market maker provides the service of immediacy, which they charge for. You are free not to use their service. You can place your own resting order, you can use an intraday cross, or a an opening or closing cross to access liquidity. You could also execute a block trade through a bank. You could use a dark pool. Or you could use some combination of all of them.

The underwriter is like a market maker only in the fact that it is taking principal risk when it purchases the shares. It typically already has buyers lined up for all the shares and it does not face risk from the price changing between when the IPO is priced and trading starts. That is not to say that the bank is not without risk. It has obligations like price support for the IPO.