I've been opting for an open source lightweight alternative to Google Analytics on my websites lately. Could this hurt my rankings or 'click-through-rate' on Google search results, even if I register my sites on Google Search Console.
Using Google Analytics (GA) or Matomo or Adobe Analytics or Piwik or any other web analytics software does not impact SEO at all. The only impact could be if the web analytics software is severely broken and ruins the user experience on your website.
The use of Google Search Console (GSC) is not related to your web analytics. There is an option to access GSC data in the GA user interface but that's it.
Using or not using GSC also has zero direct impact on your SEO.