I can't back my argument up with big o notation or such, but I don't feel like this truly compares to a binary search. Is it because I only do hard mode?
(My gut feel is unscientific, and thus my hypothesis is pretty dann dubious, but let me provide one more useless piece of anecdotal data: have you seen the wordle solver that lets you put in the final word, and it tries to guess it? I match or beat it 4 out of every 5 times. But maybe it's not the best solver?)
Maybe it's this: In hard mode, you HAVE to use the letters you've found, right?
On my first guess, I find there is an E somewhere.
Now, because I have to use that E IN MY SUBSEQUENT GUESSES, I only have four spaces per guess now, and I still don't know where the E goes.
Blah. I'm not very convincing. Maybe what it comes down to is letter combos. Like if you rule out H, you also rule out CH, SH, TH, PH, WH, and GH. Those are two-letter combos ruled out for the price of one letter. With those combos gone, the likelihood of each companion letter goes down, too! (Or if you find an H, you're statistically way more likely to have one of those combos than not. (I think.))
But E... E is ubiquitous. It can go just about anywhere, with any adjacent letter.
In short, I think this game can't be simplified to binary searches. Words have patterns that shortcut things faster than a 50/50 search on one letter. That's my hypothesis.