First off, I understand what you are looking for, and it is in no way unusual to want to have instrumentation and information about what is happening. There are many products out there, first and foremost Google Analytics.
There are also log "mining/reporting" systems available. Awstats is one of them, but it's pretty old and I haven't used it in a long time. I'm not sure how functional and up to date it is.
Here's a partial Google list of "alternatives to Awstats":
Dynatrace.
LogicMonitor.
New Relic One.
Datadog.
Sumo Logic.
Graylog.
LogDNA.
Apache log4j.
Of these I've used New Relic, and Sumo Logic in the recent past, so it just goes to show you the many commercial and non-commercial offerings in this space. One thing you sometimes need to do, to get the level of information you want, is to modify the web server log format, and sometimes to inject additional variables into the logs. Things like session id's and cookies can be added to the logs to help establish things that can't be surmised otherwise. The details of doing this are an aspect of system administration that depend on your specific webserver and hosting environment.
One specific example, would be the IP address of the request. If your server has a load balancer in front of it, the IP address of all requests will be the load balancer, and not the actual client IP, so that is an example of where you need to customize the logs in order to see what is actually going on.
There are many many products and companies out there that offer logging infrastructure. One I've used in the past, not just for webservers, but for analysis of an entire cluster is Splunk. With that said, Splunk is a pricey commercial option.
One FOSS stack that has a lot of mindshare and users is the ELK Stack, which consists of a setup of Elastic Search, Logstash and Kibana. Each piece of that stack solves a particular part of the problem that companies with sometimes large and complicated infrastructures face in getting server side analytics. You can do some reading about it here: https://logz.io/learn/complete-guide-elk-stack/ This might be the type of server based analytics system you want, and is modern, scalable and far more functional than a simple log parser/web reports system like AWStats.
Most companies use multiple different options, as each tends to have a strength and a weakness. Google Analytics has a lot of features, but of course, it depends on the client running its javascript, and thus isn't ever going to show you requests that were still processed but didn't load javascript. If there are errors or bugs in the javascript on the page, this might cause GA not to log correctly or at all. Still you want to configure and start using GA with your site, and you will find it already gives you a lot of the functionality you want, without you having to do anything within your infrastructure. In my experience companies often use a variety of different tools. Sometimes, just looking at web logs is not enough, or doesn't really help you understand something, and you need logs of multiple different services. You might need to look at graphs of webserver(s) and your database for example, to see that a problem your system was having was related to database load at a particular time, which was in turn related to some slow queries that were running tying up the database resources for a long period of time. Resources on the server itself, like available memory, amount of swap being used, and cpu load, might show you that your server is overloaded or low on disk space. There are different types of logging and monitoring you can setup, that can often provide valuable insights into issues you will never find just looking at web logs.