Hi Jeff,
Use a third-party company to survey your employees asking questions such as, "Rate your satisfaction with XYZ" options from least to most satisfied, and I definitely agree that asking questions about to improve the employee's level of satisfaction would be valuable. Using this method will eliminate anyone's knowing who responded or who said what (unless Corporate already knows that Jane wants 21 days of annual company paid vacation days because she tells everyone every chance she gets.)
As far as knowing how engaged an employee is, I would think every supervisor can tell you which employees are engaged and which are checked out. For example, if your Maintenance Supervisor walks right past a fallen gutter without noticing it; if your maintenance techs and grounds person walks right past trash on the ground without picking it up (or really, anyone on the team); if the leasing teams' closing number of leases is declining and no follow up is being down; if your Manager cannot tell you how many vacant units and who is on notice and the breakdown of the budget (such as how much has been spent on carpet this year) - then all of these are indicators your onsite personnel is not fully engaged.
In my opinion, conducting the anonymous survey is valuable as it offers each employee the ability to contribute suggestions. I would never depend on employee reviews to reveal this information. These two things may be separate. Your engaged employee may very well perform exceptionally well but may also be planning to go elsewhere, too because there could be better benefits offered, etc.