Removed the external 100k resistor and tried using the internal pullup resistor again.

Works.

void setup() {
  Serial.begin (115200);
  analogReadResolution(16);
}
void loop () {   
  pinMode(A0,INPUT_PULLUP);
  delay(0.4);    
  int c = analogRead(A0);
  Serial.print(c);
  Serial.print("\n");
  pinMode(A0,OUTPUT);
  digitalWrite(A0,LOW);
  delay(20);
}

A single press captured in all it's glory

60424
59284
59347
56480
52928
52245
50397
45113
41819
38814
35965
33451
31761
31011
30624
29825
29207
29131

So I can go back to zero external components required, probably.