There are really two parts to your question:
Here are my two cents, but as you already discover, many of them are based on opinions.
About the first question - delivering your own logs is great for small implementations - say, 1-2 servers, several databases and a full-time DBA. In environments such as lack of trap, lack of control, warning and control is not a problem. If it breaks, you do not sweat bullets, because it is relatively easy to repair. When will it break? For example, if someone accidentally deletes a transaction log backup file before it is restored to the disaster recovery server. (It happens all the time with automated processes.)
When you go beyond a couple of servers, the lack of management automation becomes a problem. You want more automated email alerts, notifications when log delivery exceeds X minutes / hours, warns when copying files takes too much time, simplifies the processing of multiple secondary servers, etc. This means that people are turning to alternative solutions.
On the second question - I will say this. I work for Quest Software, the creators of LiteSpeed, a SQL Server backup and recovery product. I regularly communicate with database administrators who use our product and other products such as Idera SQLSafe and Red Gate SQL Backup to simplify backup management. We create GUI tools to automate the log shipping process, provide you with a nice graphical panel showing exactly where your bottlenecks are, and help make sure your applications are covered when your primary data center is down. We sell a lot of licenses. :-)
If you flip your own scripts - and of course you can - you will be completely alone when your data center goes down. You will not have a support line for calling, you will not have tools that will help you, and you will not be able to tell your colleagues: "Open this graphical interface and click here to drop off." You will try to get them through T-SQL scripts in the middle of a disaster. Expert database administrators, who have a lot of time on their own, sometimes prefer to write their own scripts, and this gives you a lot of control, but you need to make sure that you have enough time to create and test them before you take the job On him.