This response is based on the information provided. We suggest you discuss with your contracting team, program manager and/or legal department as appropriate.
First, we want to commend you on your thorough and well done research.
To the specific question, " Is it possible to use FAR part 13 to award IDIQs?" The answer is yes. The more important question is, "Is it practical or useful?"
If following the rules of engagement outlined in FAR 13.106-1(a)(2)(iv), one anticipates multiple awards and is required to award to all of them. With the IDIQs in place the rules of FAR subpart 16.5, i.e., Fair Opportunities come into play. The selection process requires more administration. Each DO/TO would be under the SAT. In the BPA arrangement your can have a single contractor and even with multiple contractors getting to award and performance/delivery will routinely be quicker.
We suspect that most units do not do it not because they believe they can't but because they believe it is not as efficient.