Is it critical not to update the last page update date when related data is updated?

Question

Search Engine Optimization

Let's say we have a product category page, this category has its own description, and there are also links to product pages.
Does the sitemap.xml need to update the last update date for the product category page if the information about one of the products has changed.
The fact is that it is not very convenient to keep track of what exactly has changed when saving the product, the name and title image (which are shown on the category page) or the description of the product, which is available only for the product page.
The bottom line is, from an SEO point of view, how important is it to specify the date and time of the change for the product category page, or can you save it? After all, the reverse situation may turn out, we change the description of a product that is not on the category page, and the date and time change while the category page itself has not changed in any way.

Answer 1

From the point of view of SEO, the sitemap is far from the most important element of optimization, if the site has a clear structure, then the robots themselves are well versed with indexing / reindexing pages, and the specified dates and priority indicators for pages in the sitemap are ignored. It happens that a sitemap is not necessary at all if there are no problems with indexing all pages.
If you choose exclusively from your options, then the second one is worse - when the data changes, but the date does not. After all, if the robots updated the page index, taking into account only the date in the sitemap, you would have problems)

Answer 2

Let me rephrase the question! What is worse in terms of SEO?
1 Option - the data on the page has not changed , but the date of the last update has changed
2 Option - the data on the page has changed , but the date of the last update has not changed