In templates with TemplateData, only search the template name and description from TemplateData, not the whole doc. On a template doc can be pretty much anything, so in my eyes it makes the most sense to consider only the information within TemplateData.
The entire docu should only be searched as a fallback option for templates without TemplateData.
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This ticket is a duplicate of T274906: Prioritize TemplateData description when indexing/searching. It was declined after extensive investigation. While something like this is technically possible (with the most likely approach being a custom CirrusSearch profile), the investment is significant.
I will keep this open for now. Maybe it's worth having the discussion again at a later point in time.