After much playing around with your PowerShell script... it's all very odd.

Observed Behaviour

Firstly, the Open method on the Workbooks object only reports 14 parameters when $excel.Workbooks.Open.Invoke.ToString() is run. The ouptut reads:

Workbook Open (string, Variant, Variant, Variant, Variant, Variant, Variant, Variant, Variant, Variant, Variant, Variant, Variant, Variant, Variant)

However, there is some kind of overload method for 15 parameters because, when using the more verbose error read-out $Error[0]|format-list -force i did two test calls, the first with 15 parameters and the second with 16.

15 Parameters

Exception             : System.Runtime.InteropServices.COMException (0x800A03EC): Unable to get the Open property of the Workbooks class                           at System.Management.Automation.Interpreter.MethodInfoCallInstruction.InvokeInstance(Object instance, Object[] args)                           at System.Management.Automation.Interpreter.DynamicInstructionN.Run(InterpretedFrame frame)                           at System.Management.Automation.Interpreter.EnterTryCatchFinallyInstruction.Run(InterpretedFrame frame)

16 Parameters

Exception             : System.Management.Automation.MethodException: Cannot find an overload for "Open" and the argument count: "16" --->                        System.Reflection.TargetParameterCountException: Cannot find an overload for "Open" and the argument count: "16"                           --- End of inner exception stack trace ---                           at System.Management.Automation.ExceptionHandlingOps.CheckActionPreference(FunctionContext funcContext, Exception                        exception)                           at System.Management.Automation.Interpreter.ActionCallInstruction`2.Run(InterpretedFrame frame)                           at System.Management.Automation.Interpreter.EnterTryCatchFinallyInstruction.Run(InterpretedFrame frame)                           at System.Management.Automation.Interpreter.EnterTryCatchFinallyInstruction.Run(InterpretedFrame frame)

As you can read above, the method does support 15 parameters but not 16. However, no matter what values are supplied to the 15th parameter, it will fail to open the file.

For the record, it works with 14 or less arguments, it is only the 15th argument that throws the exception.

Conclusion

From everything that I've seen, I can only conclude that there is a problem with the Excel COM interop support in powershell. Supplying the 15th parameter as $missing simply should not change behaviour according to the Workbooks.Open() reference. There being a problem with Powershell's Excel COM support is further reinforced because, when the VBA script is run as a macro, everything works as per the documentation.

Suggested Next Steps

The first work-around that comes to mind is to write the VBA script as a macro and store it in an excel file that is used specifically for running VBA scripts from the command line. It would take minimal effort to implement and it is known by testing from both the OP and myself that it will work.

If you have any difficulty triggering the macro from powershell, see Calling Excel macros from PowerShell with arguments