Almost every SaaS log management tools will allow you to do this very easily.
Logentries uses the concept of a log set and log for aggregation and search.
You could create a log set for all of the web server logs. Your colleague could create a separate log set for all the SysAdmin logs. Our agents can do this for you automatically. The free version will handle aggregation and search of data.
These videos will walk through the features and capabilities of Logentries
If you are interested in a live demo send me a direct message and I'll connect you with one of our engineers.
I saw my Tradfri gateway connecting to "webhook.logentries.com" .
A quick look at the website ( https://logentries.com/ ) it seems a service for log analytics.
I've used https://logentries.com/ for general purpose logging including some NSErrors. I've never thought to try logging all NSErrors since the system uses them a lot and I don't want logs full of noise though I guess you could if you wanted to by swizzling the NSError init methods.
At this point because you don't know your use case and sounds like you're still trying to figure out, you're probably better off paying for a solution. https://logentries.com/ or https://www.sumologic.com/
I've been using https://logentries.com/ for this. I basically just replace the NSLog with a log to their site. It's free for my use and they have an iOS library that makes setup pretty easy. Their log browser is a bit slow but otherwise I've had no problems.
Have you looked into Logentries? It's similar to Sumo Logic, but offers a few more benefits, like natural search (as opposed to a query related search), and they offers unlimited logging, rather than paying per GB, so you can send all logs in any format, from any source, from any system - then only pay based on the logs you use.
No idea in regards to patch management for Windows (In a previous life the Windows team used WSUS and/or Altiris, but can't vouch for it, I come from the Linux side of things)
For log centralization, a team mate is testing this here at the moment, https://logentries.com , and seems quite happy with it so far.