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An XML sitemap is a mere pleasantry. Google will index your site with or without it, provided that every page is accessible via normal hyperlinks.
Google crawl behaviour is not predictable. But it makes sense that if Google has trouble visiting your site, it may not see+follow its links, and may not achieve a full crawl. Repeated server problems may result in Googlebot not bothering to come back.
And it has been recently noted that slow download times DO affect your ranking.
Concentrate on fixing the server problem and abolishing those timeouts. That is your #1 priority - don't worry about XML sitemaps until you have that problem licked.