Of course all hardware clocks should be set UTC, but there's a lot of benefit from using the locale to understand what's going on in relation to the human viewing the log.
If i'm a human waking up at 3AM and I look at the logs for this server, the first thing i'm going to think is "is the time on this server the same as where I am?" followed by "how long ago did these events occur in relation to me?" The easiest way for any human to do this is to compare timezones. If you condition yourself to know the UTC difference of every time zone, then this works automatically, but [i'd argue] most humans are better at estimating timezone difference from other timezones, such as how California and New York are 3 hours difference, and New York and London is 5 hours difference.
In terms of correlating log events across the globe, you really, really want a log parser and correlating tool. They make your life easier and reduce time and complication during events. Splunk, Loggly, Graylog, Logstash, ELSA, etc. Don't look at your logs by hand. You'll be sitting there all day with 6 split windows looking at mysql, nginx, app logs, kernel logs, mail logs, security logs, blah blah blah, just on one server. When you horizontally scale your app, whether it's 2 or 2,000 servers, you need something to parse and correlate your logs so you can say "show me all logs from 11:00PM to 2:00AM from Web Cluster B", you get to save 10 minutes on your outage, and you don't miss anything.