Investigators found that the TCAS of the Legacy was not operating and could not correspond with the 737’s system.
Not quite right. The TRANSPONDER of the Legacy was inoperative (because it seems the TCAS was switched off), hence that lack of a TCAS warning in the 737.
Lack of a TCAS only means that no CA guidance will be given to the pilot of the VLJ. Assuming that aircraft above 5,700kg have a TCAS and the VLJ has a transponder, then at least one aircraft will be given CA guidance.
That said, it does seem ridiculous to me that a high-performance jet of any type, especially a single-pilot one in busy European airspace, does not have TCAS.