As it turns out, the server it did install just fine on had .Net 3.5 installed on it as well as .Net 4.6. Where as, the one the install failed did not. Compare the one below (successful) or the pic above (failed).
So, I installed .Net 3.5 on the server, and then the Engineers Toolset install worked just fine. I guess maybe the error message should read 3.5, not 4.0.