By chance, I work on a project that does just that with 18,000 network elements: -)
I do not think you'll find a product that solves this for you.
What I believe to be the way for you in this case is to periodically run scripts that copy these logs to you for a local machine, and then automate your analysis of these logs yourself.
If the machines you manage are more or less of the same type, it will be a very easy service. In my case, the work is quite complicated because there are hundreds of different types of machines, some running operating systems that I had never even heard of ;-)