WordPress or your own stack?
The question is not which system is better. It is who maintains the site in three years, and what it has cost by then.
The case for WordPress
You can find someone who knows it anywhere. There is an extension for almost everything. And if your agency disappears, the next one is up to speed quickly. That is a real argument, and anyone who ignores it is selling you something.
The case against
Those same extensions are the most common source of security holes, of slow pages, and of the complaint that something stopped working after the last update. A site with thirty plugins has thirty vendors shipping changes independently of each other.
When a page builder is enough
When the site is a business card, rarely changes, and nobody needs to find you through Google. For a club or a side project that is a sensible choice, and any agency selling you something else in that situation is selling too much.
When custom becomes cheaper
As soon as the site is supposed to earn money. Then load time, findability and how quickly content can change start to count. Custom does not mean more expensive: expensive is every agency starting from zero. Reusing the same proven foundation delivers faster and for less.
The question that settles it
What happens if we part ways. Can you get your content? Does the site keep running? Can somebody else take it over? An agency without a clear answer is selling dependency, whatever the system.