Korban Pesach Conundrums

There is a well-known line we recite in the beginning of the Haggadah in the paragraph of Ha Lachma Anya. “Anyone who wants, may come and eat; anyone who wants, may come and join in the Pesach.” Seemingly, this line is a throwback from when we had the Korban Pesach. In the times of the Beis Hamikdosh, we used to invite any interested parties to partake of the Korban Pesach. Nowadays, when we sadly no longer have the Korban Pesach, we still say the same text.
However, this theory could not possibly be true. To partake of the Korban Pesach, you must have been one of its subscribers before it was slaughtered (Pesachim 61a). It is too late on Pesach night to decide that you want to partake of a particular Korban Pesach. The Korban Pesach was already sacrificed on Erev Pesach.
Indeed, this actually explains a rather peculiar custom that developed in certain communities. The custom was not to have guests for any meal the first two days of Pesach. Furthermore, they altered the text of the aforementioned line to read, “Whoever wants can come but may not eat anything.” This actually could be a throwback from when we had the Korban Pesach. You couldn’t invite last minute guests to partake of the Korban Pesach. Therefore, even if someone came and visited, they could observe but not eat.