Wedding Invitation Formality

I wouldn't say letterpress is necessary.

I am admittedly a sucker for letterpress, and even if it was a couple that I wasn't extremely close to (like I was an invited friend, but not a family member or bridesmaid-level friend) I would be inclined to keep it simply because I like letterpress!

That said, I think a nice paper and envelopes with beautiful calligraphy would go much farther in giving guests the "feel" you're going for. The real first impression will be the envelopes - I always know from an envelope if I am receiving something special.

For me...

A non-white envelope (unless white is truly the predominant color in the event, in which case, see next point.) I once received this deep, rich burgundy envelope with gold calligraphy. It sticks with me to this day as one of the most special things I've pulled out of my mailbox.

In the case of a white envelope, a really luxe paper, and a non-black ink choice. (See next point.)

Hand calligraphy, definitely, and preferably in an unexpected color. In the case of a darker envelope, a metallic like gold or silver. In the case of a lighter envelope, something dark but not black - like navy on a white envelope, or emerald on a light sage green envelope.

Names in a very beautiful script, with the addresses in a more block font. You can have the entire address in an ornate hand, but I feel the mixed hands make the names feel more special, being the only complex script.

What shade of grey? Is it a light or dark shade? How dominant is grey in your wedding colors?

There are several different ways this could be done.

The first - ivory envelopes. Ivory is not quite white, and there is something very sophisticated about ivory. A calligrapher could certainly find you an iridescent black (have a bluish sheen) or a charcoal grey ink for the addressing, if you want to keep in that neutral palette. Alternatively, if you have a third non-neutral color in your palette, the address ink could be in that color. There's nothing saying you have to matchy-matchy the address ink to your invitation text.

The second - grey envelopes, in particular if grey is a dominant color in your wedding. You could go with either silver (if it's a darker grey) or an iridescent black again (if it's lighter). Gold on grey is also nice if you are using gold anywhere.

You have a lot of unique options in that color realm, since ivory and grey are both neutral.