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I bought an old Raleigh for £20, stripped it down, serviced all the bearings, removed the tacky decals, cleaned it up and reassembled in an afternoon. The only things I had to replace were the tyres and front inner brake cable which I had laying around in the garage anyway.
It's a comfy as fuck rigid cromo mtb that does all the things the OP wants and more and cost less than the price of a single chainring
