In the case of the OPC and Westminster, the historical answer is very simple: Machen believed that it was interference from the PCUSA denomination that brought down Old Princeton and made WTS necessary. A similar case could be made for Anglican Theological colleges in England. At the very least that highlights the fact that denominational control is not always a positive feature.
In most cases, the seminaries that are under denominational control tend to be small (both the denominations and the seminaries). The advantage for smaller denominations is that it makes it easier to preserve your distinctives; the challenge is to resource a seminary adequately with excellent faculty and with the financial resources that are necessary (faculty pay, books, buildings, etc). Sometimes a denominational seminary is not as attractive an option for students outside that denomination, which makes it harder to make it viable. As a result, sometimes these seminaries rely on part-time faculty who are primarily pastors rather than being able to hire men who are able to devote their full time to teaching, research and writing. There can be some advantages to that model, but there are disadvantages as well.
Independent seminaries can draw faculty from a wider base than if a denomination insists that all faculty members have to belong to that entity. WTS and WSCAL. for example, have faculty from the OPC, PCA, ARP, IPC (UK) and URC churches, and have had presidents from the PCA, OPC and URC. It's also easier for them to draw students from a wider base, without being a significant financial burden on denominational funds (the ARP sends serveral hundred thousand dollars each year to Erskine College and Seminary; I imagine that the PCA sends far more to Covenant).
Moreover, it's not obvious that everyone in the OPC would agree on which seminary should be "theirs": WTS? WSCAL? Greenville? Not everyone in the PCA is thrilled with Covenant, nor does everyone in the ARP feel that investment in Erskine is the best use of limited finances. Yet changing a denominational seminary is a near-impossible task. Personally, I'd much rather the ARP took the money we currently send to Erskine and allocated it to the Presbyteries to fund scholarships for men under care to attend schools which those presbyteries have approved. That would provide real and immediate accountability.
In some respects, independent seminaries are far more accountable to their constituents than denominational seminaries, since they have to make the case to every student and every donor as to why they should invest there. They can attract a broader range of students and donors, providing greater resources for training than smaller schools. If you think the independent school has gone off the rails, it is easier to cut off their influence in your denomination: just don't send them students or dollars.
The bottom line is that there is no one "Biblical" model. Every model has its strengths and weaknesses in a fallen world.