Rav Avigdor Miller on Bris Invitations
Q:
Why don’t we invite people to a bris milah, if we invite them to other simchas like a wedding, a pidyon ha’ben and so on?

A:
It’s a custom not to invite to a bris because they might not come; and to snub an invitation to a bris is a serious thing. That’s because a bris is a demonstration that you don’t find in a wedding, or a pidyon haben or anywhere else. A bris is a demonstration of the relationship between us and Hashem, of the covenant between us and Hakodosh Boruch Hu – that’s what it is. And therefore if somebody invites you, you’d better come.

And therefore the minhag is, better not to invite people because you’re making trouble for them; it’s very serious — Eliyahu Hanavi is snubbed if you don’t come to his party. That’s how important a bris is – it’s a very big celebration of the the first covenant between Avraham and Hakodosh Boruch Hu and it’s a bris olam – it’s an eternal covenant. And therefore if you’re invited, it’s just too bad – you have to show up and participate in this great demonstration.