Why can no weather forecast reach beyond about two weeks?

Because the atmosphere amplifies its own measurement error. Edward Lorenz found it in 1961 by accident: restarting a weather simulation from a printout rounded to three decimal places instead of the six the machine held, he expected a near-identical run and got a completely different forecast within simulated weeks. The difference between 0.506127 and 0.506 had taken over.

That is the defining property of a chaotic system - not randomness, but exquisite sensitivity to starting conditions in something entirely deterministic. Since we can never know the state of the whole atmosphere exactly, the tiny gaps between our measurements and reality double roughly every few days until the forecast is worthless. Better instruments push the horizon out slowly and cannot remove it: halving the error buys a few more hours, not another week.

So forecasting changed shape instead. Modern practice runs the model dozens of times from slightly different starting states and looks at how far the results diverge - which is where a percentage chance of rain comes from. It is not a statement about how much of your area will get wet, but about how many runs of the model produced rain where you are. The two-week ceiling is a property of the atmosphere rather than of the computers, and no amount of processing power will move it.