I don't think so, because both losses were due to bad management decisions under irrational political pressure, not any lack of engineering knowledge that more unmanned testing could have provided.

Challenger was lost because NASA ignored a critical flight risk with the SRB joint O-rings. And by "ignored", I mean "documented that the risk existed, that it could result in loss of vehicle and loss of lives of the crew, and then waived the risk so the Shuttle could keep flying instead of being grounded until the issue was fixed". They didn't need more unmanned testing to find the issue; they needed to stop ignoring it. But that was politically unacceptable since it would have meant grounding the Shuttle until the issue was fixed.

Columbia was lost because NASA ignored the risks of tile damage due to their belief that it couldn't be fixed anyway once the Shuttle was in orbit. But that meant NASA also devoted no effort to eliminating the risk of tile damage by fixing the issue that caused it. Which again would have been politically unacceptable since it would have meant grounding the Shuttle until the issue was fixed.

Testing wasn't really the issue with the loss of the two shuttles. In both cases, it was mostly a management issue. For Challenger NASA had seen o-ring erosion in earlier launches, and decided it was not a big risk to the crew. Then they launched Challenger against the recommendations of the engineers in charge of o-ring seals. For Columbia, they has seen foam strikes in earlier launches, but since they had not caused catastrophe in the past, they decided that foam strikes were acceptable. Even when it was clear that a large foam strike had occurred on the launch of Columbia, management wasn't concerned enough to try to get ground-based images of the shuttle to check for damage. Could Columbia's crew have been saved had they known the extent of the damage? No one can say of course, but not even trying to do everything possible was inexcusable.

They very nearly lost the first shuttle they launched. Jumping straight into manned testing was quite reckless, but politically necessary. If they had tested the shuttle without crew, that would have gotten people thinking that crews probably aren't necessary for a lot of shuttle missions, in particular launching satellites. It also would have prompted people to compare the cost of shuttle launches to other unmanned rocket launches, in particular for commercial satellite launches (which they were doing until the Challenger disaster.) These are comparisons that would have been very problematic for NASA as a political entity.

The problem there is the Shuttle was deliberately designed so it couldn’t be flown unmanned, which risked lives and wasted money for lots of simple satellite launches.