First of all this site isn't running a custom CMS. F5 Sports Media is a small, niche player in that market for sure but it's running on a few other sites too (probably most notably TexAgs where it seems to have been developed). I have no idea if it is FOSS, but I doubt it. My guess is they provide customization and maintenance for a subscription fee. For the features this site has a FOSS CMS or even a custom full-stack JS site would be pretty easy to do, but who needs that when a passable set and forget alternative is within your budget. Sure they could probably save some money there (just one example they're paying for SSL in 2021) but how much?
Second Cal is already "partnered" with BI in that they provide direct content from time to time. See articles with "Cal Athletics" credited on the byline.
Third, hosting is cheap now, even on Azure which they are using (although there are cheaper alternatives). The major cost for this site is payroll.
The problem with a site like this is and will always be that there's very limited scalability. There is no other viable business model other than "passion project" or labor of love" or whatever else you want to call it. The revenue potential just isn't there, at least not without significant innovation to create new streams. That's why the big corporate sites cover all major conferences. That's also why there's never been a true competitor for this site despite its long history of, shall we say, technological limitations. Nobody's ever gonna make any real money covering only one school, particularly one like Cal, so why bother? There are probably a number of things they could do to generate more revenue, but they're higher growing fruit not without cost or risk so there's little incentive. As it is I appreciate this site for what it provides and don't dwell too much on what it doesn't. But I disagree that it's improved since the ownership change or that it's (still) the best Cal site.