Perhaps I'm not educated in this subject, but if vowel harmony means "all the vowels in a word to be members of the same subclass" then does this mean that English has vowel harmony too? For instance, words like lambaste, parka, almost, also, dollar/scholar/colour (AUS/UK English), eerie, collage/montage, follow/swallow/hollow, finish, folklore, borrow, moron, ardor, although, diminish, etc....use the same vowel class in their pronunciation or spelling. Isn't that a "vowel harmony", in a way? Or am I missing something?
English doesn't have vowel harmony.
"Vowel harmony" refers to situations where there is some process that changes vowels to be in the same class as other vowels in the word, and/or there is a constraint against having vowels of different classes in a word.
You can see examples of vowel harmony processes in Turkish on e.g. this web page: Vowel Harmony (some examples: the plural of kedi is kediler, the plural of kuş is kuşlar).
Processes like this may be explained in terms of a "constraint"; vowel harmony constraints often also seem to show up separately in base vocabulary e.g. there are few native Finnish roots that contain both front and back vowels. In many (perhaps most? I don't know) languages with vowel harmony, this constraint is violable and "disharmonic" words with vowels from conflicting classes do exist. In particular, it seems compound words are rarely subject to vowel harmony constraints (they aren't in Finnish or in Turkish), and loanwords may not be subject to vowel harmony constraints. But there's a difference between having a violable constraint, and not having any apparent constraint at all.
This is just a general summary. I am not an expert, and even experts still have much to learn about the specifics of what vowel harmony is. Here are some more detailed explanations and discussions:
"Vowel Harmony: Statistical Methods for Linguistic Analysis", Rebecca Knowles
Formal and Cognitive Restrictions on Vowel Harmony, Sara Finley (Google Books)
Effects of contrast recoverability on the typology of harmony systems, Gunnar Ólafur Hansson
English doesn't have any processes or constraints like this (as far as I know) so it does not have vowel harmony.
There aren't any English suffixes that use different vowels depending on the vowels in preceding syllables, and there aren't any general restrictions based on vowel class of which vowels can co-exist in an English word.
The suffix -y found in messy is pronounced the same in the words foamy, woody, warty, hearty which all have different vowels.
Vowel harmony is a property of a language, not of a word or a collection of words. It's simply the way the term is used. The examples in your question do each have all vowels from the same class, but the term vowel harmony is not used to describe this.
As the other answers aptly explain, English doesn't have vowel harmony.