You are correct that there has not been time to do a thorough study of the differences between natural infection and the response to a vaccine. It is not crazy

You are correct that there has not been time to do a thorough study of the differences between natural infection and the response to a vaccine. It is not crazy to want to wait a bit if you were recently infected. However, there are good reasons to think that vaccines (particularly the mRNA-based ones) have a longer lasting protection than that a natural infection. They include only a very small part of the spike protein; with the booster, the force the body to create lots of targeted "memory" cells that are designed to last a long time. A natural infection may (i) target part of the virus that changes faster than the spike protein and (ii) without the booster effect, not produce as many memory B-cells.