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There are some shared vocabulary, but thats pretty much it. Kurdish is much closer to Farsi.
Back in the 90's, it was not that hard to find Turkish people saying exactly this.
This was before most had the Internet, so they could say stuff and get away with it. The idea was to belittle the Kurds, and deny them their identity. Obviously, not serious scholarship.
"Indo-Anatolian" for the higher-level family is a term that some may use here and there, but no expert would ever make a "well, acktually..." remark as you just did to those in a discussion, professional or layman, using "Indo-European"
Despite the terms used, Anatolian and non-Anatolian languages are seen as two separate branches.
Maybe I'm being too charitable, but I hadn't read that user's comment as a "well, acktually", especially given the chain of comments, but perhaps I'm wrong.
Certainly, that Hittite broke off first is the consensus, and thus it makes sense (but isn't the consensus) that we should say something like "Indo-Hittite". BUT, one is a matter of describing history, and the other is an arbitrary label attached to that description.
Your "well akshually" was a nice way of softening the original poster's brief comment. I just don't think any of us truly disagree (barring that user's subsequent thoughts, of course).
In any case, my point was the early split of Anatolian and "the rest", however you want to call it.