Despite 8.5mppa, I think Bristol has a relatively small, and fragile, demand for business flying, mainly due to the relative proximity of BHX and LHR.
bmi had an awkward size fleet, of awkward sized expensive to operate aircraft. I believe most of their routes ex-BRS were commercially viable, and I tend to think it was their other endeavours (particularly operating routes ex-FRA and ex-MUC as LH *codeshares*, not wet leases) which dragged on them.
An example to illustrate the problem of aircraft size. Ex-LHR (ex-MAN, ex-BHX), LH operate 319/320 aircraft, and there's usually a decent availability in the K L T buckets, which are affordable flying. There's usually very good availability in V W S buckets, which is pricier, but still within range. You'd have to be unlucky to find an LH flight ex-UK with only Q H U M, because that's where it gets painful. If you end up with only Y B avails, forget it. You're better off walking.
bmi was almost never able to offer K L T. If you got lucky, you might find some W avails. More often than not, I'd be up in Q U. On a short-haul fare, that's about a £200 difference on a round trip. On a long-haul fare (UK-SFO) that's about £600-£800 difference on a round-trip. If you breathe very deeply, you can justify spending an extra £200 to fly ex-BRS over going to LHR. That is much much harder to do when it's £800. And even harder again when the difference is closer to £1,000 because if you're going to have to go to LHR, you have a wider choice of airlines, some (most) of whom will be cheaper than LH Group.
So if you can't find an aircraft that can offer decent inventory (which I'm guessing is about 90 seats, as KLM do well ex-BRS), your potential passengers will just go to LHR. I suspect that isn't true of some of the other regional destinations that the LH Regional fleet serves - it's their price or no price.
You're quite right to say it is anomalous that our only alliance/hub-carrier is KL (with a little nod to EI). It's possible that until economic ~90 seat aircraft come back in-vogue at LH Group (more CRJ900s? A220-100s?), we may be waiting another 5 years for connections to the world via anywhere other than AMS.
M_V - you've reminded me of the trips I took on BACon's MUC-BRS route, which was a weird MUC-BRS-GLA route. Everyone staying on-board for GLA, had to get off, bus to the terminal, walk through passport control, go back upstairs and through security, and re-board again. An absolute palaver.