My guess is still something with the two new DNS protections in Anti-bot. Unfortunately, granular control of these features is not exposed in R81.20, but it is possible to toggle these two DNS protections in R82 individually without completely disabling Anti-bot. Also see the discussion of "dns=bg" here: sk92224: Optimizing the categorization of DNS traffic by changing the Resource Classification Mode, ...
The fact that websites perform fine for 60 minutes after the initial delay in access, and then the initial delay returns after 60 minutes again, screams some kind of cache issue to me, possibly CRL/OCSP, but it seems too early in the connection for that. Are you using FQDN Domain objects in your Access Control policy or HTTPS Inspection policy? The cache timer for domain entries is 60 minutes, so I'm wondering if when that expires, there is a long delay having to retrieve the FQDN again and get it in the cache, but then once the site's domain gets into the cache, life for that site is good for 60 minutes, until it times out of the cache and the gateway has to retrieve it again, which causes the delay.
Is DNS traffic subject to Geo Policy or Geo Updatable object blocking for any countries? If Cloudflare's initial servers are located in a prohibited country it may take awhile before your DNS resolver hits upon a server that is in an allowed country and not blocked.
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