I personally haven’t found this to be the case. I’ve often found that starting out with an apology leads the offended to be much more receptive to what you have to say next. Best case scenario the person you are apologizing too isn’t offended at all and tells you so. Worst case scenario the offended uses your apology as justification that a wrong was committed in the first place. Thankfully I have only ever encountered either the former or people somewhere in the middle.
Opening up with "I'm sorry" sets a tone for the conversation. If my last interaction with someone was a little tense, I need to _reset_ the emotional balance before moving forward.
In this hypothetical, I don't believe I did anything explicitly wrong. But I also didn't handle the situation well. Maybe the "sorry" is "sorry for not addressing your concern". Or maybe it's just "sorry that 'mansplaining' is a situation you encounter frequently". It doesn't have to be BLAME. Just empathy.
All an apology costs you is pride, which is basically free. If it gets you to a place where you can have a normal conversation, who cares who has done what wrong? The point of talking things out isn't about assigning blame.