With a reasonably sensitive contact pressure sensor, you can pick up a sensor value just after having made initial contact (so a small measurement error before the sensor is sure it has contact) and then continue to move the probe down until the sensor reaches the force limit.

An issue here with an automatic device is that if there is a problem with the contact pressure sensor, the system will continue to push down so there must be a second set of mechanics that limits the amount of force available. That is very very hard to do when you have multiple sensor probes concurrently pressed to the skin unless each probe both have a contact pressure sensor for detection and some engage mechanics that have hard limits for how hard the pin may move forward (like a spring or similar). Your individual pins are very small, so even very low forces can do damage.

Another issue is that multiple contact pressure sensors will not be identically sensitive. So in soft tissue, there can probably be significant differences in how far you can press the individual probe pins before the system decides the required force has been reached. So there must be a way to calibrate each sensor tip.

One option for the mechanics would be pneumatics. With a fixed air pressure used to drive all measurement pins forward, you know exactly what force each sensor pin may push forward with before reaching equilibrium. So you wouldn't even need a contact pressure sensor to decide when to stop - just some sensor to decide when the measurement pin makes initial contact with the skin. And of course a sensor that measures how far it has been pushed out by the pneumatics. The skin contact could potentially be capacitive when it's enough to just know "touches skin" and the need to know how hard has been removed.

But still consider the manual route - one trick used there often is to project an optical grid on a surface. An operator can very quickly press a probe to each grid point. It isn't unreasonable that an operator with a single probe may be just as fast as an automatic contraption with 20 concurrent probes just because of the dexterity of our hands and our ability to quickly switch between high movement speed and great precision.