Been through that whole spectrum of emotions in a similar position over the years and, ultimately, came out at the same conclusion as you have. There is very much still a place for a dedicated "ad-hoc request" BI team in the modern mix of data and analytics support. Tightening up your work reception processes to weed out the timewasters early on will help reduce your stress levels over the longer term. But I would never advocate retiring the function altogether and give in to the creeping hell of self-service BI.
Looking forward to seeing how this pans out for you over the next 6-12 months.
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