My organization wants to do Quality but does not want to be ISO (Construction)

@Jorge Beluche
I've seen that a lot. Allow me to tell a quick story.
My first experience with quality was in a Mechanical Inspection education institution. They were certified in 2004, renewed every cycle flawlessly, until 2012. On that year, the local economy was greatly affected by the price of the Crude Oil Prices and our main customers were in the Oil & Gas industry. Two of the main refineries in the country had to be shut down and one, that was close to an inauguration, never got to its debute.
Our costs remained close to what they always were (normal to a company that sells services), but our revenue dropped significantly.
Most of our customers were now from different sector, none of which require a certified QMS. So, we continued with all Quality procedures, but without paying for the ISO 9001 audits and maintenance.

The Message is: If you don't have a requirement stating that you need ISO 9001, you can very well implement a control system that will make your management much easier using PDCA, Continuous improvement and clear KPIs.

A good end to the story: In 2017, when I left the company to move to Canada, the company saw a positive trend in revenue again, not thanks to the Oil sector, but renewable energy. As they kept the QA system despite the requirements, they called back the certifiers and re-instated the certification.

Now, if the problem is implementing because the your colleagues don't value the advantages of the certification, then they can say they WANT IT, but they think it would look nice to have a sticker to glue on their truck's tail gate.

I hope this inspires you to go for it, and not the second case. Good luck!

There is nothing stopping you from using ISO 9001 as a good foundation for your quality management system. You can conform without going through the certification process. Once you have everything in place, maybe your organization will see the advantaged of certification.