Where is your "RemoveExistingProducts" action sequenced?  If it is at the end of the InstallExecute sequence (which is the Wise template default) try moving it to between InstallValidate and InstallInitialize, as this positioning removes the old app files before installation of the new ones, and avoids the sort of problem you are experiencing, which is usually caused by not running the UpgradeSync process.

Unfortuntally that did not help.  I've been working on this upgrade for several days and it has been very frustrating.

I've been very careful when creating prevouis upgrades to run UpgradeSync. 

I'm not sure what to do next?

Generating a verbose log of the upgrade process is the best way forward. You can then examine the sequence of events and find why the files are not being installed after being removed.
Is a reboot required after the upgrade by the way?

I did generate a log. I can see where the files are removed, but there is nothing else in the log about the file that were removed.

No reboot is required.

I've been using this installation for over 2 years. I have added over 20 upgrades to this installation without issue. There is no way to copy a file that guarantess it will be installed?

There has to be a reason for your install suddenly misbehaving in this way.
Are the new versions of the files correctly versioned internally, so that they have a clearly incremented file version?
Where exactly are the files removed when you have RemoveExistingProducts sequenced between InstallValidate and InstallInitialize?  They should be removed by the RemoveExistingProducts action.
Then there is nothing to stop the replacement files being installed - UNLESS for some reason their feature level is set to a higher figure than the INSTALLLEVEL property, for example, or unless there is something else flawed in the installation that is preventing the new files being installed.