Doesn't need to be a single person, but yes, all good products come from someone or a (generally) small team, spotting something, and fixing it.

As for the marketing department - depends on definition, but assuming you mean (as it mostly is nowdays) comms, it's two things:

1. Making sure that potential future customers know you exist, so when they enter the market to buy, they know you're relevant and can purchase from you.

2. Making sure that when someone is in the market to buy now, that they're more likely to buy from you, because you have a compelling reason for people to pick you, and not the other brands in the space that they can think of.

Mostly 1, some of 2, if the marketing team is good. Ratios vary as to how much though by more than you'd think, depending on industry, customer lifecycle, brand maturity and so on.