Q: Once my car reaches 60MPH, the steering wheel/vehicle shakes and vibrates intensely.

Tires that "appear" perfectly balanced on a dynamic balancer can still vibrate like crazy on a car if there is significant radial force variation in the tire construction, defective belts, cupping, uneven wear and so forth. If you have moved the tires to different positions though and the vibration is essentially unchanged that reduces, but doesn’t eliminate, the chance that you have a tire problem. Replacement wheels that have a different offset are not recommended as such alters the suspension geometry and creates loads that are not contemplated when the car was designed. If there are ANY loose suspension parts, yes, that can set the stage for vibration because any rotating element such as a tire/wheel assembly has a residual imbalance. Normally, that residual imbalance is invisible to the operator because the suspension tightly holds the rotating element rigidly in an axis. But, if the rigidity is lost, due to a loose ball joint, the imbalance can appear. Normally, the fact that a wheel alignment was done means that, at that time, the suspension was checked for looseness simply because it is impossible to align a car to factory specs with ANY (even one) worn suspension part. Of course, that’s a big "if", so yes you have to inspect the suspension because an alignment could have been attempted without bothering to check the condition of the suspension. In your circumstance, to resolve this thoroughly inspect the suspension looking for faulty parts, including wheel bearings, struts, and shock absorbers. If the tires display ANY cupping and any obvious tread defect, mount new tires (preferably Michelin or comparable which we know are made such that radial force variation is minimized) on OEM wheels using a radial force variation balancer, confirm that imbalance is low (less than 20 or 25 pounds radial force) and re-test the car. If the vibration remains, that means there is an issue with other elements such as drive axles, motor mounts, and so forth. A certified front end specialist from YourMechanic can further evaluate this for you during a front end inspection.