Are Unicode characters in passwords really useful?

Unicode characters are supposed increase the difficulty of guessing passwords. The reasoning is that they expand the search space for passwords from the 62 possible alpha-numeric characters up to 1,114,112 total possibilities for each Unicode character.

Doing the maths

Let's assume that we're using upper and lower case ASCII passwords (52 possibilities per character) vs. Unicode. According to Wikipedia: Unicode defines a codespace of 1,114,112 code points in the range 0_hex to 10FFFF_hex. Let's say that about 100 of them are special control characters that can't be used, leaving 1,114,012.
ASCII ==> 52 characters. (upper and lower case)
Unicode ===> 1114012 --> log₅₂(1,114,012) = 3.5238
i.e. Every random Unicode character has as much entropy as ~3.5 random ASCII characters.
...I hope that's correct.

Side note about passphrases

There are only about 10,000 or so English words that people regularly use. That makes every word not 52ⁿ bits of entropy for n characters but one token from a pool of 10,000. Every word gives you an equivalent of using 2.33 random letters. If the number is 100,000 each word then gives you 2.91 random letters per word. I like to think I'm a smart guy, but I'm certain that I don't know anywhere near 100,000 words.

Just type out the Unicode character code

Here is an idea. When you use a Unicode character you need to type out the decimal/hex code. If we allow for all upper+lower+numerical characters it gives 62 bits per character, but when you are pressing each key to input the Unicode character you're only adding 16 bits per key-press. You have to remember and type the hex characters anyway.
e.g. If you want your password to be "m💩nkey" (If that didn't work for you it's "monkey" with the 'o' replaced with a pile of poop.) you need to type this sequence of letters: "m 1 F 4 A 9 n k e y". (I think in Windows you press Alt+ the decimal rather than hex characters, but it really doesn't matter.) That pile of poop added 1,114,012 bits of entropy (let's not assume that a brute force is going to prioritise poop over other possible Unicode characters). If you typed out the characters "1F4A9" to your password, assuming 62 possibilities per character, you would have added 916,132,832 bits of entropy. Especially if for whatever reason Unicode is not allowed in passwords it seems to me that there would be no harm in just typing out the characters if you are otherwise going to force yourself to remember a hex string.