If you don't need disks, my advice is to look at bikes a few years old. If you do that, 3K will buy you a LOT of bike. And someone else already put the first scratch on it for you. Local classifieds may yield some interesting bikes to look at and now you don't have the shopping restrictions. Note that bikes are HOT right now. I had a track bike / fixie I wasn't using and sold it much quicker last month than I would have imagined.
I did some shopping in 2016 and got a lot of advice at the time here too. My budget was about 3k and instead of getting a mid level new bike I ended up with a C50 Colnago, full Campy Record set for half my budget. It has been a great bike and a lot more fun to own than a run of the mill machine. Yes, I probably could be faster (?) with a new bike, but not at that price point. And I am not racing, just riding for fun.
G
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