I should have phrased that a bit more clearly: it's perfectly possible for different people(or even the same people, with different bitcoins) to have 'eh, you'll just have to trust us on that; but it sure is fast!' and mathemagic-crypto-money at the same time, it's not as though the presence of one activity irrevocably taints the entire pool of bitcoins for the other; but any specific bitcoin (or subdivision of one) can only have one or the other:
Someone always has to be the one holding the private keys, and they are the only ones for which the elegant math and its assurances apply(and even for them, it only applies to the BTC side of a transaction, not to whatever the other party promised to deliver in exchange for that irreversible payment).
You can layer all sorts of arrangements on top of that, as the exchanges do(where the exchange, not its users, is the one holding the keys; but near-instant clearance, escrow, transaction reversals, etc. are all possible because all 'transactions' are just the exchange shuffling bits internally), or a transaction broker/insurer type model, as you propose, or a conventional escrow service, or any number of other possible contractual arrangements.
Nothing about bitcoins magically prevents the use of any of the zillions of arrangements we've applied to commodities and mediums of exchange throughout history, my point is just that, if you are a participant in those things, the properties attributed to bitcoins are largely irrelevant to you(basically the only one that matters is double-spending prevention, without which inflation would render all of them worthless within a few hundreds of milliseconds), and if you do want to enjoy all the properties of bitcoins, the list of things you can do with them gets a great deal shorter.
That's not some scathing condemnation, every medium of exchange in history has only functioned because some semblance of law, custom, or a mixture of the two kept the surrounding institutions vaguely in check, fraud within sufferable limits, and so on; but bitcoin brings nothing new to the table here, except in the very narrowest range of uses, where its mathematical properties do hold, and if you wander outside of that, it's open season.