Former tenured professor.
I think your impression is generally spot-on although there's nuances and it's more complicated than that.
In general, universities are suffering from many of the same pressures as other fields. Nonprofits are being run as profit-generating centers, and the implicit promotion track goes from faculty to senior faculty to administration, if you're even on tenure track. The administration salaries are increasing, as are the number of administrative positions (deans, associate deans, vice associate deans, assistant deans, etc.), and the structures are becoming increasingly hierarchical. Things are very top-heavy and expensive to run because the salary budget is so disproportionately distributed.
I don't think that's all of it, though. The rest of it is harder to quantify, but is the flip side of the educational bubble coin. Students are rushing to go to college in record numbers, which then puts pressure on colleges to stand out to attract more students and more high-paying students who become high-donating alumni. Institutions that should be getting public funding are not, which then trickles down to students; other private institutions then benefit from the increased tuition norms etc.
It's a complicated problem that involves HR and administrative practices and norms, economic incentives within university funding structures, incentives coming from a broken employment system in the US, problematic incentives coming from federal grant structures, etc. etc. etc. etc.