Tim, the problem with your assumption is that any offer made will most certainly be made with inspection or other contingencies. Not sure how it works in NC, but in Texas (where the buyer is located by the way), an agent having a buyer make multiple offers without any contingency (assuming the buyer is not prepared to purchase several properties) would border on malpractice.
I've been making multiple offers for more than 13 years and thus far have never had an issue, except that I can lose the fee associated with my termination option on the homes I choose not to purchase after an offer is accepted. However, at a hundred bucks or so I consider this the cost of doing business.
I understand traditional agents/brokers not be actively investing may not agree with this, but nonetheless, this is a very common practice in Texas...which again is where the poster is located. Since the poster is investing, it is not only silly and unproductive to wait for individual banks to respond (many banks state upfront they may not ever respond to your offer unless it's accepted), it is a sure fire way to ensure you fail as an investor.