Desirability of chaining an LCG in an authentication scheme
Would it be too slow to make for a good authentication mode?


No, it would be too insecure for a good authentication mode.

In particular, given any message $M$ that's at least two blocks long, it's trivial to find a second message $M' \ne M$ that has the same tag (no matter what the key is); and hence one could be substituted for the other without detection.