I have been working as a DBA for about 8 years (roughly) and I don't see the "light" at the end of the tunnel for becoming a Sr. DBA anywhere near. Every time I think I learned enough that I can make things work as I need, I hear about a cool new technology.
I agree that there isn't a "finish line" for being a Sr. DBA; it depends entirely on what you, the individual, wants and enjoys. Also, what job duties you enjoy. Some companies consider the DBA role to be a "can you do the thing with the database? then it is a DBA role". So BI work falls on the DBA; data warehousing falls on the DBA; backup and maintenance falls on the DBA; windows updates to the SQL servers falls on the DBA; database object creation falls on the DBA. Other companies have more segregated duties. Data warehouse falls on the database analyst. BI work falls on the BI team. windows updates falls on the infrastructure team. Object creation falls on the database developer. Backup and maintenance falls on the DBA.
Some people really enjoy the back end index tweaking and statistics maintenance tasks, installing updates... all the stuff that happens to keep the lights on that nobody other than the DBA really think about. Others like their work to be flashy and in your face so they may enjoy building the SSRS reports. Others enjoy the development work. I think a lot of it falls on what you enjoy doing as well.
Your career path aiming for a Sr. DBA role may be all about building janky reports while mine may be about getting that index tuned JUST right so the inserts are optimal while still having good performance on the selects. Or you may enjoy grabbing data from all over the place and making a nice little data warehouse or master data management system while I prefer building my own backup and maintenance scripts.
And the DBA role is changing. With the advance of Azure and getting the data into the cloud (either via VM or SQL Server in Azure), and the cool new features in newer SQL versions (like the query store), the duties of a DBA are changing. If you want to be a Sr. DBA at the end of your career, you need to keep up with the new technology.