Money aside, do you think that you'd like the building process or do you just want to ride?
The thing about building a bike slowly as you acquire parts is knowing when it's done. That's definitely my M.O., but the bikes tend to continually evolve over time so they're never really "done". I'm constantly thinking of something I want to add or change or do to it. I guess it would be more expensive to build a bike that way if you insist on all brand new parts, but a used bike will have all used parts too. Over time, your collection of bike components tends to grow and you can do a lot of putzing around for little or no money. My grand son and I built up a pretty nice road bike with leftover parts my son and I had and and I think the only things bought new were tires and handlebar tape.