To attach the balls, apply the solder past to the pads on the PCB, place the parts, then preheat the balls to just above the reflow temperature of the solder paste. Place the balls on the pads. They should melt the solder. Then put the whole thing in the oven and let the rest of the solder reflow.

Heh heh heh...
Now try doing that with case-hardened steel balls. Without disintegrating the PCB in the process.
It is much easier if you can get some gold-plated steel balls - finding them and paying for them might be more difficult though.

I used gold-plated rods in place of the 6 balls, and gold-plated ball-ends in place of the 3 rods, for some of my creations. But I happened to have such parts left over from other R&D work (not from my budget!).

On the other hand, you could try the jewellery channels:
https://www.ebay.com/itm/3Pair-Set-G...4AAOSwZQRYc9X3
But a caution here: the gold 'plating' on all these jewellery bits is extremely thin and when you try to solder to it the gold 'plating' just melts off the steel surface. You would have to go for a mechanical connection, which can be a bit unreliable, or spot-weld to the shaft. They are stainless steel, not case-hardened, but with low forces they can work.

Your next problem will be the stability of the PCB material. That is only medium. It can be subject to slow creep during use. Using a ceramic replacement material is more stable, but you have to find the ceramic and be able to machine it (and pay for it). Macor, which is a machinable glass, is a possibility here IF you have the budget. But it is really NOT cheap!

Sad experience shows (me) that one spends more time realigning the probe tip (every time) than actually using it.On the other hand, one can make a Z-axis probe with micron resolution quite easily, and one can equally make a single axis single-direction sideways probe without too much hassle.

Does all this mean I (and many others) have spent ages experimenting with making 3D probes? How did you guess? Fwiiw, I have 3 - 4 3D probes of various sizes and several Z probes and Z-touch plates on the shelf (and more bits too). They work, very well too, but they are also too much bother in practice.

Cheers
Roger