The most critical one of these I have run across is senior management bypassing the agreed process.
Implementing a product development process is a evolutionary stage of organizational development. You don't just put one in "because", you put one in to address emerging problems in the organization, which is typically conflict of priorities over what limited resources should be working on, resulting in project delays, budget overruns, off-spec deliveries, and miss communication of what is being done.
Understand that most companies start with 2 functions, the product factory (engineering) and sales. The senior management teams fulfills the product management function, and makes all priority calls. As the company grows and both product lines and segments serve expand, there is just too much going on for senior management to make "shoot from the hip" calls. Some of the above issues start to emerge, and there is a need for a structured process.
The first step I address in establishing a NPD is establishment of the "steering committee" which is a cross-functional leadership team, that reviews proposals, makes the selection for investment, reviews progress, and makes the resource priorities calls and trade-offs. The cross-functional and leadership make-up is important. Once you get this in place, then the NPD process is about standardizing and streamline what information is presented, how, staging the work, etc. There are numerous approaches available here, but the first requirement is creation of this "steering committee".
The challenge question then to the founders, and typically senior executives is whether or not they are ready to give up control of product decisions to the "steering committee" and agree that any pet projects must go through the committee? If they answer is NO, my position is "Thank you for the opportunity to discuss your needs, but I can't help you" and I walk.
I have learned over 30 years of doing this, that this type of change cannot succeed without full senior leadership buy in and commitment.