Not all BCBS plans are nonprofit actually. And most comparisons I can find for nonprofit medical facilities show lower costs. I haven’t found many studies on pure on profit health insurance vs for profit insurance, but I did find a Harvard paper which compared specifically BCBS plans that converted from non profit to for profit, and here’s an excerpt from that:

Looping back to the theoretical models of NFP and FP health care organizations, the findings are consistent with models in which NFPs prioritize enrollment over profits (equivalently, models in which FPs prioritize profits over enrollment). While theoretically this difference in emphasis might not manifest in higher premiums or lower quality because FPs could be more efficient and find it optimal to maintain substantially the same premiums and quality as NFPs (and still reap higher profits via lower operating costs and/or medical expenses), empirically we do find there is a tradeoff: consumers face higher premiums when large NFPs convert to FP status. Although we do not directly study quality, we find no indirect evidence of quality improvements, as inferred from a model of employee healthplan choice. Moreover, we do find evidence that rivals of converting plans experienced sizeable increases in medical spending following conversion, a result that suggests FPs are likelier than NPs to engage in risk selection practices (e.g., denying or deterring enrollment of individuals with poor health or high health risk, a practice that was legal during the study period).

Here NP is nonprofit, FP is for profit, and NFP is not for profit. Bold emphasis is mine. You can read the study here:

https://www.hbs.edu/ris/Publication Files/20130370_manuscript_c83842eb-f97b-4c84-b356-c72d163dff9b.pdf

So I would find actually the opposite of what you said, in aggregate, according to this study. Secondly, I still argue for expanded Medicaid and a public option / single payer. I’ve worked with large population datasets from US and internationally — invariably the health outcomes and monitoring, quality of data and followup, are all better for single payer systems.