I also wonder whether the reason the "natural vs vaccine" immunity question may not have a great, direct answer is it wasn't asked until covid-19.
When I was asked about this by a 'vaccine skeptic', I said having the vaccine after an infection was clearly a huge benefit over having just the vaccine or just the infection - so do it. I've also seen a study where the super rare, problematic responses to the vaccine seemed to be the same symptoms someone would experience with an infection sans the vaccine - only the effect would be dramatically worse with an infection. (Sorry I can't offer sources at the moment - moments from sleep.)
When I was asked about this by a 'vaccine skeptic', I said having the vaccine after an infection was clearly a huge benefit over having just the vaccine or just the infection - so do it. I've also seen a study where the super rare, problematic responses to the vaccine seemed to be the same symptoms someone would experience with an infection sans the vaccine - only the effect would be dramatically worse with an infection. (Sorry I can't offer sources at the moment - moments from sleep.)