Using a catalytic converter with the exhaust heat recirculation system VS without

When is it appropriate to use a catalytic converter that does/doesn't have the "exhaust heat recirculation system"?

Examples might include :

• Vehicle shipped with (or without), why deviate from the plan
  
• Measurable loss of MPG
• They cost the same, and it doesn't matter either way
  
• Global warming is real
• Just another part that fails, give it up and save a water pump

Images for either design on the first page : https://attachments.priuschat.com/attachment-files/2018/09/153912_2010_Toyota_Prius_Repair_Manual_-_Exhaust.pdf