How can this be a "mistake"? They created something that looks like a captcha to fool the client into believing it's an actual captcha. If they did't know how to make a proper captcha it's better to tell the client so someone else does it.
What if they don't know they don't know how to make a captcha? It's an obfuscated image, those are easy to make and check! If you don't know basic things like "never trust the client", and you don't know that they exist to know, then you may not know to tell the client to have someone else do it.
That doesn't excuse the programmer. As a web programmer, it is, to some extent, their job to know when they're out of their league. But second-order knowledge can be a rare skill.
Yes, this exactly. Donald Rumsfeld got no end of flack for his comment (distilled here) "there are known knowns, known unknowns, and unknown unknowns", but it's actually a great statement - in this case, there are some people who know they know how to make captchas, some people who know they don't, and some people who don't know that they don't know.
Many programmers have no idea how a CAPTCHA is supposed to work. It never occurs to them to think though how someone would break it. Someone tells them the client wants a CAPTCHA, they go "oh yeah, that's those weird letters on the screen", and are probably pretty proud of how they did it.
Don't believe me?
Think about how often you see obvious SQL injection problems - the same (lack of!) thought process is responsible for both.
You are assuming that the client knows what a CAPTCHA is. Probably the manager at the client-side said "Oh yeah, before I forget to mention it, add that funny image you see on websites - you know, the CAPTCHA thing, a guy at my gym said it improves security. We definitely want good security in this project!".
