Nice, you discovered Chemandy. Good calculators there.
Yes, fat traces. As you've discovered, even with ground fill, trace width is on par with substrate thickness. The real winner is thinner substrate, whether 40 mils (or thinner) 2-layer stock, or 4-layer (inner ground plane with usually 10-15 mils of substrate on top). Then you can connect without necking.
And, impedance being what it is: you can compensate for short necks, to some extent, by following that with an overly fat trace of about the same length. (Which, in turn, can be followed by..) Now, the precise geometry necessary to fully compensate the impedance discontinuity -- up to the maximum possible flatness * bandwidth -- is nontrivial to calculate (you're building a lowpass filter, actually), but for hand-waving purposes, it's enough to note this. So, as a filter, you're adding series inductance (necked down trace), which causes a bandwidth reduction (and consequent power mismatch, if the filter bandwidth is cutting into the signal bandwidth). A parallel capacitance (fat trace) can "peak" that up, by about a factor of 2.
Or you can simply use, say, 100 ohm traces on board, and add impedance matching networks wherever you need to connect with the outside world! (This might not be a beginner option, as most appnotes and components are intended for 50 ohm systems, and adapting components may be tricky.)
Tim