When using a point-point simple rigid link, I get a fatal error that a solid element has singular mapping. That seems understandable, I think.
The issue is that solid element do not have rotational DOF. Given the thin nature of your geometry, you can use shell elements, which have all 6 DOF; using an Advanced Rigid Link will allow your to connect the two points while also releasing the rotation DOF (but still preventing rigid motion of the link). I've made modifications to you CAD model that reflect the best way to go about this (I deleted some geometry and replaced it with surfaces to resolve a Hard Point mesh control issue).
For some reason, the modal results were the exact same with and without a very stiff spring, but it seems like the rigid link is a better solution to pursue.
Not surprising. A rigid link and an infinitely stiff spring are the same. Even though the spring you've define has finite stiffness, it's most likely much large that the local stiffness associated with the elements it's connected to. You're also only looking at 5 modes and your analysis is set to Quick Check (which means it solves the entire model at a uniformp= 3 and does not escalate any element orders; I suggest running anMPAanalysis with maxp= 9 and 1% convergence). Also keep in mind that the lower frequencies are more accurate if a higher number of modes are included.
With all that out of the way, you should also consider whether it's realistic for connect the two parts together via two points. It's almost certainly more appropriate to connect them via two small surfaces. If you do need to do this, you can use an Advanced Spring or a beam element along with either rigid links or weighted links to capture the kinematic nature of the connect. If you run the model I uploaded you'll notice it's hard to get the first mode to converge; this is due to the point-to-point connection (which looks like it's converging to something in the low 6 Hz range). If you were to make a small surface-to-surface (using the method mentioned) you'd get better results. Even a small surface (0.2" diameter) causes a significant change in the first mode (around 29 Hz).
EDIT: I'm having issues uploading the CAD model. If you private message me an email address I can send it to you.