Yes, everyone, including coders, thinks they are the best product designers. The difference is that top-level coders actually have the influence to implement their ideas, whether they're good or bad. That's how you get your foot in the door. Walking in to a startup wanting to be the one that tells everybody else what ideas they should be implementing is not going to get you far. I'm not saying that product managers are not useful, but good product managers are data driven and their main job is to gather and summarize the available data and enable a conversation within the company about the product direction, not to be the grand visionary that hands down ideas from on top.