What are the most important product management skills or perspectives that others inside an organization could benefit from that would improve their day to day?

Your team size, overall organization dynamic, and product maturity all shape the skills you need to improve day-to-day operations. Distilling to the basics, I'd go with the following:

• Customer Focus: PMs who put the customer at the center of their decision-making process often help other departments understand their customers' needs better. 
• Communication: being a skilled communicator, both in writing and verbally, is key to enabling cross-functional teams.
• Agile: Many PMs use agile methodology to manage their projects. Understanding agile principles and developing your own "secret sauce" would help you rally the teams forward. 

Great question. I would frame this as more "Product skills and mindsets" vs. just Product Management as I see these skills and mindsets across the community of Product roles. Also, I'm going to do what a good Product Person would do and prioritize for what I think would have an outsized impact on the whole company.

Accountability to Outcomes over Outputs

A product trio cannot survive long if they don't deliver outcomes and they just keep touting "look what we shipped." A sales person cannot survive long touting all the demos they made without posting actual sales. There are certain parts of the org that operate like this naturally because its high visibility and it gets measured, so the opportunity is to get the whole org operating like this. A great example is that in my recent history an internal team rolled out a new tool for employees. There was a ton of fanfare about the launch, but months and months later adoption was abysmal. If this was a Product team that launched this it would have been a huge miss in terms of product market fit, yet this was generally seen as a success because the narrative only focused on delivery, not outcomes. What happens here is that you are doing your org culture a disservice. When some teams are accountable to outcomes and others outputs, you cant "be all in this together."

Demo Your Work

First, note that this can be an enabler to helping achieve the point above. In general modern Product teams follow some sort of Agile, and in there should be the aspect of doing sprint demos. As a reminder we demo work to get feedback and keep the shared understanding train on the tracks. My experience is that the outcomes are always better when you show your work early and often to customers - you learn more and get more access to signals to increase your odds of hitting that outcome you are driving for. The odd thing here is that this feels a uniquely "Product" behavior, which it doesn't need to be. Imagine the impact of the Sales Engineer demoing the new demo system to their customers (sales team) as they are working? Imaging the FP&A team demoing the new process to do annual budgets to their customers (department heads) as they are working?

Test Your Assumptions

I observe that people make lists and talk about assumptions all the time, but don't really know what to do other than curate the list and reference it at the RCA meeting :) Product has to constantly balance speed with risk, and this is a real solid technique to help a Trio out. The core concept here is use assumption mapping to get a clear view on assumptions, then do simple quick tests (not research projects) to reduce your high risk assumptions. The reason I think this is valuable to the whole org is because 1) we all could use a better way to de-risk our work 2) most people don't know what to do after they create the list of assumptions. Either people slow down to assess everything or get overwhelmed and just YOLO it.

There are 3 main skills that product managers are known for that I think ALL members of an organization can benefit from:

• Customer empathy

• Communication

• Collaboration

Customer Empathy:
While this is a key skill all product managers should have - it is something that EVERYONE in the organization can benefit from. Ensuring that the company is thinking about the customer from all directions translates into a more customer centric company. And solving for the customer almost always translates into higher brand trust - one of the key ways to drive long term business value.

Communication:

In my experience, 90% of the challenges I face at work (and probably at home!) come back to communication. Communication is key in planning, expectation setting, alignment, strategy, process, and so on. From finance to customer service to product to engineering - ensuring there is alignment and expectations are set - is key to a high functioning team. All of this is dependent on high quality communication.

Collaboration:

Product managers know collaboration well - since they cannot deliver a product or customer value without their partners in design, engineering, and so on. Their success is 100% dependent on their ability to work with others to deliver a unified product experience. While this is true for product managers, it is true for most roles. if you have employees at your company, that means there is a group of people trying to get something done together. And without collaboration skills, this becomes much harder.