Try 470Ω resistor with anything between 0.1uF and10nF capacitor for the RC filter. 10K is too much that´s why your waveform is way slow. What you see at the output of the 74HC is also normal?
You stay for too long in the "unknown" region. For slow rising signals you need gates with schmitt-trigger inputs so that you do not get this problem. And yes the capacitor is required as it is what makes the smoothing.
Thanks for giving me these hints. I have never drawn a connection between resistor values and rise times in circuits. To me the resistor value was mostly determined based on the acceptable current in my circuit (i.e. do I want to afford 10 or 20 micro amps when the button is pressed). I took your suggestions and made some more measurements, comparing different resistor values. For those interested, here are my results:
470 Ohm, no Capacitor
1 kOhm, no Capacitor
10 kOhm, no Capacitor
100 kOhm, no Capacitor
I have so much to learn... I think I'm slowly rediscovering RC circuits from first principles.
Edit: the 74HC14 is a gate with a schmitt-trigger input. Are you sure your single gate ( I suppose ) logic you use there has a schmitt trigger input?
Edit No.2: You use the SN74LVC1G04 which is not a schmitt-trigger input gate. It is not noise what you see, it is the IC not knowing in what state it should be. Not the best long term for its operation probably also.
To be absolutely honest, I was looking for "74HC" and single input and was mostly ignoring everything else, thinking that "oh yeah, that 74HC series ... I guess those chips are all the same, except for the number of inputs?" and did not bother too much with schmitt triggers, inverter, hex inverter or other terms. So it's very possible that I bought the wrong chips.

All I really cared for was a sharp edge, no matter if it's rising, sinking, is inverted or whatever. Just give me my damn warm fuzzy digital behaviour :-D