Your thoughts on who should be invited to a wedding are spot on. Thirty-six years ago, my wife and I decided that only people who were important in our lives would be invited to our wedding. To ensure this would happen, we paid for everything ourselves.
We told our two sons about what we did before both of them got married, and they thought it was a wonderful idea. Throughout the planning stages of the two weddings, we constantly heard "I can't wait for this person to meet that person." At each respective wedding, I was asked by my daughters-in-law to offer a toast. Here's what I said on both occasions:
"Please raise your glass. I'd like all of you to look around the room. You might now know a lot of the people here, but everyone that you see has played a significant role in the lives of the bride and groom. You are, to a large degree, responsible for the great people they are today. So, before we toast them, lift your glass and acknowledge each other. You've done good work!"
Afterward, as at our wedding long ago, people who did not know each other beforehand introduced themselves and shared stories of how they had supported or influenced the bride and groom. The weddings were, as you suggest, "solemn" occasions, but they were also joyous ones that everyone still talks about today.