@jaykar: You'd have to build the whole "default" utm value logic that is in Google Analytics's "backend" into the browser (=Tag Management System) via JavaScript, thus you'd need the whole logic regarding what is organic search, what is referral traffic etc. in your browser's JS. You'd spend some time on this and then you'll have to maintain it. Is this complexity really necessary?

As for forwarding the actual utm parameters of a campaign (not "google / organic" bc you'll never see this in a URL) to links on the same page, that is possible with some JavaScript, but I would make sure your website developers are aware of that as changing actual links on a web page is not something you usually want to do with a Tag Managemenent System because it actually changes the user experience and how the webshop works.

I try to stick to the rule that a TMS should only do things that do not change the user experience/website functionality. Because that's not my ballpark and I don't want to take the risk of my TMS breaking any website functionality -> creates mistrust from development / product management side.