I saw a post by Brent Ozar the other day, and it was a bit disturbing to me. A survey he posted on Twitter showed most people don't bother with error handling in T-SQL. I'm not surprised, though I wonder if people think about the "rare" as including all the one single statement queries they write. I certainly don't expect error handling for simple queries when they are a single statement used to return data to a client.
However, in stored procedures and multi-statement batches, I see the same thing as the results Brent published. Very little error handling. I've tried in my demos to add it and be sure that I set a good example, even though most of the time I'm showing code that just always works. I know what data I'll enter for a demo, and I don't run into issues.
Many years ago, decades actually, when I was in college and early in my career, I saw many application software packages written without a lot of error handling. This included C/C++ at the time, which seems crazy. In the last decade, I've seen a lot more robust error handling (and testing) added to the work of many application developers. It's the request that my professors and more than a few bosses always made, but in the late 80s/early 90s, few people actually followed through with. In the 2010s, this seems to be more common, and not surprisingly, software quality has improved.
In many ways, database developers are less mature than application software developers in many ways. Less error handling, less automated testing, and that's understandable. We haven't had great tools or patterns to help us easily adopt these practices as a habit. In addition, the way in which code is compiled and then executed doesn't make this easy for us. The platform and structure of the language create complexity that isn't present in application languages.
If you haven't written this in the past, as Brent notes, there's not need to worry about going back now. Either your code is working or not working, and if it's the former, no reason to revisit it. However, I might ask that you experiment with error handling for new code. Learn how to use these structures to protect against your users entering strange data. Something they are likely to do.