These types of questions probably cause me more stress than any other. Being asked to think of a situation where you felt a certain emotion or had a certain mental experience is not actually that easy for everyone. My memory is terrible at that kind of thing.
It's like my memory lacks a necessary index for what you are asking. I can start going through every major work experience I've had in sequence and filter out the ones that fit, but we might be sitting there in silence for 2-3 minutes which gets awkward enough in itself.
And it's not an inability to remember mistakes specifically, I mean the old "tell me about a time where you solved a problem with no prior knowledge of the domain..." gets me. I solve problems like this ALL THE TIME. That's a skill I have. Fetching those times from memory is not a skill I have.
Luckily I've had these questions in enough interviews that I generally try to have some examples in mind.
How will I know if this person is good at, say, sales or biz dev in a vertical with long lead times? I can't vet that skill set directly; I can only ask the candidate about it. Behavioral questions give me something to cross-reference in the absence of on-the-spot performance data. So do case questions, which (at least ostensibly) tell me how a candidate thinks, and how she organizes her thoughts. (And of course, domain-knowledge questions help me assess the depth and quality of a candidate's experience in the field.)
Behaviorals aren't perfect, though. They can be gamed. It's not too much of a stretch for a candidate to memorize a half-dozen anecdotes to trot out in response to any question that begins with "Tell me about a time when..." (Nevertheless, you'd be surprised by how few people do that sort of prep work.) Canned responses are especially common with candidates straight out of school or grad school, where they are often trained in how to answer behavioral and case questions.
From my experiences with employers in many parts of Europe, they value partial contributions to teams more than complete individual achievements. Talking to much in "I did..." sentences instead of "We did... (...and I helped with...)" can come across as self-centered, which often is considered a negative trait here.
"Tell me about a time in which you..." works very well. If they resist answering the made mistakes questions we prompted them to understand that an honest answer is important. If they could not do that then they were not considered.
These were not developer roles but most were very technical / analytical roles for an organization that paid well and candidates knew up front they would be interviewed this way. In addition some jobs were "cased" where you had to solve a problem and talk about how you solved it to the interviewer. Great fun when I went through the two days of interviews. I ended up doing maybe 200+ behavioral interviews over 6+ years for the company. I saw this approach work time and time again.
Some people could not answer for themselves: "the team did..." "They did...". When pressed we usually discovered that the candidate did not have the ability to get the job done themselves.