The loss of the Challenger was the 25th manned orbital mission. So we can expect that it might have taken 25 unmanned missions to cause a similar loss of vehicle. But what would those 25 unmanned missions have been doing? There just wasn't 25 unmanned missions' worth of things to find out. That's also far more unmanned missions than were flown on any previous NASA space program before manned flights began.

Even leaving the above aside, if it would have been politically possible to even fly that many unmanned missions, it would have been politically possible to ground the Shuttle even after manned missions started based on the obvious signs of problems with the SRB joint O-rings. There were, IIRC, at least a dozen previous manned flights which showed issues. There were also good critiques of the design available at the time--which, in the kind of political environment you're imagining, would have been listened to. That design might not even have made it into the final Shuttle when it was flown.

In short, I don't see your alternative scenario as plausible, because the very things that would have been required to make it possible would also have made it unnecessary.