Hi cyrilrueg,
The idea (of the tutorial you shared) is that the models (e.g., hair and beards) are based on a universal head (Base Head).
To fit this to the target head, the Warp application
https://faceform.com/download-wrap/
is used to change the shape so the objects fit the new head.
This is done by estimating the surface location among the closest vertices to each "card" and moving the card's axis to this Point. (At least, that is my understanding of the process.)
If that is not a given, a manual adjustment is needed, and it might be some work to get a good match. Not fun. So, let me share an idea here.
I do not have the needed Objects here, but I would subdivide a copy of the target head so the "resolution" would be high enough and then use the System Preset in XPresso "Get Nearest Point" to plant the card axis to this Point.
I presume the model (Hair/Beard) is merged, so the single cards must be created to run an iteration for all and then merged again afterward.
This is not something I like to do here, as it is a guess, but here is a little example: Move the "Search" Null, and the "Found" Null will jump to the nearest Point. This is the idea in a Nutshell. Instead of interpolating (as in the tutorial), we produce much more points and use those.
CV4_2024_drs_24_MOhb_01.c4d
All the best