I think OP's basic premise is mistaken (Here are about 5,700 results from Google Books for "I have wanted to ask", so it's certainly not true that we never say it.
And in some contexts ("I've always wanted to ask", or "I have wanted to ask for some time now"), Present Perfect is arguably more natural than Simple Past "I wanted to ask".
As OP suspects (and as backed up by his comment re 2,120,000 GB hits for "I wanted to ask"), the Simple Past form superficially seems almost 40 times more common.
Whilst I agree Present Perfect really is less common than Simple Past in OP's "polite question" contexts, the preference is nowhere near as marked as those figures suggest, since most of the 2,120,000 instances reference past time in a "narrative" context (where only Simple Past works).
But even allowing for that, I'm sure there's still a preference. I don't disagree with Barrie's point about Past Perfect Continuous (I have been wanting to ask) often displacing Present Perfect (I have wanted to ask), but there's at least one other factor in play here...
Both Past Perfect Continuous and Present Perfect imply strong links to the present moment. But in OP's primary context, "I wanted to ask [you] [some question] is often just a fairly meaningless "deferential introduction to an interruption" (a bit like the polite throat-clearing "Ahem...").
In such situations, the speaker is deliberately trying to create "distance" between himself/his words and the "present moment" (that's why we say "I wanted to ask" rather than "I want to ask" in the first place!). Obviously it would be counter-productive to use a verb form that's specifically adapted to linking events in the past to the present moment.
The last thing I want is to hurt you
This means that the speaker is doing (or is going to do) something that involves a risk of hurting the listener, but is stating explicitly that that is not something he wants to do—quite the contrary, he is very reluctant to hurt the listener. Crucially, the listener has not yet been hurt.
The last thing I wanted was to hurt you
In this version, the past tense, the speaker knows that he has already hurt the listener. He is now trying to atone for this by saying that the listener being inadvertently hurt was not what the speaker wanted—it was an entirely unwanted side effect of whatever it is the speaker did.
In other words, there is nothing strange about these two phrases: one is in the present tense and deals with a present-future outcome, while the other is in the past tense and deals with a past (= already known/existing) outcome.
Best Answer
All 3 options (I wanted, I want to, and I would like to) can be used.
"Wanted" is past tense, so your concern with "wanted" is valid. Good catch! The other 2 options are certainly the better ones. However, if you already sent the message using "wanted," the typical recipient of that email should understand your invite still currently stands, especially if you conclude that message with celebration details.
"I want" and "I would like" are very similar choices, so feel free to use either.
I Want:
I would Like:
(Dictionary.com)