There's a common belief in Judaism that you shouldn't formally invite someone to a bris because it obligates them to perform a mitzvah, and -- if they can't come -- you're basically setting them up to fail at performing a mitzvah.
After a bit of crazy Googling and asking around, I ended up arriving at the source of the minhag, which is the Rema, also known as Rabbi Moses Isserles who lived in the 16th century and who is known for his inline commentary on the Shulchan Aruch. I listened to a very informative podcast by YU on the topic, and it seems that our minhag derives from a misunderstanding of the text.
The Rema writes (Yoreh De’ah 265:12) that anyone who does not participate in the festive meal that accompanies a bris is viewed as if he is “excommunicated from Heaven," adding that if offensive people are participating in such a meal, one is not obligated to join them.
A basic reading of this suggests that if you go to a bris and decide to scoot out before the festive meal or attend the festive meal and don't eat anything, that you're a pretty rotten person. There are thus two aspects to the bris that play into this minhag -- the attendance of the mitzvah of bris and the festive meal that accompanies it. It's the latter that seems to be the point of contention for the Rema, not the bris itself, and not the invitation itself.
At the same time, there's an opinion that if there is already 10 adult Jewish men at the meal, the guest is not obligated to take part because the commandment will be fulfilled without him (Otzar Ha-Bris, p. 163).
So I can understand why people think that outright inviting someone to a bris would automatically obligate them to come and attend the festive meal, but ultimately it's about the decision someone makes once they're at the bris/festive meal, not before hand.
Do you hold to this minhag?