Do cameras have a mode where they take a picture, check if highlights are correctly exposed and then adjust the settings?

Kind of. The mode is called 'automatic exposure'.

Your mistake is to consider blown highlights or clipping as 'unreliability' in the light metering, which in most situations is not the case. Except for under the most tricky lightning conditions, any halfway decent camera has a light meter very well capable of realizing already before taking the picture, that the chosen exposure settings will cause clipping, either in the highlights, in the shadows or both and to make a deliberate choice somewhere inbetween to minimize clipping.

If the scene you are trying to photograph has a higher dynamic range than what the sensor is able to capture, the camera must make a best effort choice and guess where to cave in, clip the image and lose details in highlights or shadows. This decision may not be what the photographer finds best, but that is what you have exposure compensation for.

If the camera had just recognized clipped highlights and reduced the exposure to avoid clipping on the high side of the image, the adjustment would have increased clipping on the low side and caused you to lose more shadow details and that may also not be what the photographer wants. The point is, that it does not really matter if the camera makes the decision before taking the picture (as when using automatic exposure) or if the camera tries to improve a botched exposure by adjusting the settings and taking a new picture automatically (as you seem to want). In both cases, the camera must compromise and choose between loss of highlight or shadow details.