You could win the tour de france doing the road stages on a medium quality bike. (if you put good tires on it). Also given your price range, some of the objectively best bikes (meets weight minimum, good aerodynamics) are in the $5k range, and the more expensive ones are worse but have big name value!
That's definitely not true...at the level of the TDF, every gram of excess weight matters, and there's a vast weight difference between a cheap $5k bike and a "fancy" $20k bike.
If you're not at the TDF level, there's no point in spending more than $2k on a bike since the difference in weight won't matter.
It is true because weight is limited by the UCI limits. The bikes are so light now some riders have had to resort to adding extra weights to their bikes to pass the UCI inspection. The best rider on a $5k or $20k bike will still be in the mix regardless. I'd say they could even get away with a $2k bike.
It's not so much the weight but aerodynamics. The top aerobikes (not timetrial bikes) use about 10-20% less power than a non aerobikes at a speed of 45km/h (which is not a high speed during racing). That has actually been the big innovation in the last 5 years, the realisation that aero is more important than weight, except for the most extreme mountain stages. All the big manufacturers are now making aerodynamically optimized bikes.
Actually the rules restrict minimum weight and they often have to add ballast. And reducing weight does have diminished returns at some point, particularly on flatter sections. I think the weight obsession is more aimed at selling to amateurs than making pros faster.
