ANSWER: While Exodus 12:43-49 sounds "unyielding," we must understand it and the New Testament scriptures in context.
The key verse from which most have concluded that "no uncircumcised man can do Passover" is verse 48:
If a foreigner staying with you wants to observe Adonai's Pesach, all his males must be circumcised. Then he may take part and observe it; he will be like a citizen of the land. But no uncircumcised person is to eat it. (CJB)
But let's look at that verse.
In Hebrew, Exodus 12:48 permits an understanding that the "circumcised male" means "of the circumcised", that is, YHWH's people. Even in Moshe's time, Hebrews were known as "the circumcised" because of the long-held sign of the covenant with Abraham (Genesis 17:11). So in the verse where it reads, in English: "let all his males be circumcised" and "let no uncircumcised eat of it", it is not just referring to the physically circumcised. (Bear with me). Looking at the first part of the verse, it reads (in English): "And when a stranger sojourns with you and shall perform the Passover to YHWH, let all his males be circumcised" (JPS). But that English only conveys what it conveys...in English. Indeed, it conveys only the message of "physically circumcised".
But a more accurate translation of the verse to English, taking care to capture what the original Hebrew conveyed, would be: "And when a stranger sojourns with you and shall perform the Passover to YHWH, let him be of the circumcised", meaning "only let a stranger partake in Passover who has taken Me (YHWH) as his Elohim." The same is true of the last part of the verse: "But let no uncircumcised eat of it", would be more accurately written as: "But let no man who does not have YHWH as his Elohim eat of it."
There is a very compelling reason to believe that the Hebrew permits this interpretation. If we accept only the narrow interpretation conveyed by the English, i.e., that the verse only addresses physical circumcision, then a stranger who was physically circumcised could partake in Passover even though he actually worshiped another god! We do not believe that of those goy in the "mixed masses" who followed the Hebrews out of Egypt, (or their descendants at the end of the 40 years), would have been permitted to partake in the Passover after they crossed the Jordan if they had received physical circumcision but had not abandoned their Egyptian gods!
Much later, in Jeremiah, something new about circumcision is revealed supporting the "YHWH's people" interpretation of Exodus 12:
Jeremiah 9: 25 "The days are coming," says ADONAI, "when I will punish all those who have been circumcised in their uncircumcision - 26 Egypt, Y'hudah, Edom, the people of 'Amon and Mo'av, and all those living in the desert who cut the edges [of their beard]: "For although all the Goyim are uncircumcised, all the house of Isra'el have uncircumcised hearts."
Here we learn that YHWH is expecting more than the outward, visible, physical circumcision. He is expecting that the circumcision be a sign of your heart; that your obedience be honest and toward seeking righteousness. (See Jeremiah 9:13-14.)
Now fast forward to the time of Paul. He writes in Colossians 2:11:
"In Him [Yeshua] you were also circumcised with a circumcision not made with hands, in the putting off of the body of the sins of the flesh, by the circumcision of Messiah," (ISR).
And further in Romans 2:24-29:
"...as it says in the Tanakh, 'For it is because of you that God's name is blasphemed by the Goyim.' 25 For circumcision is indeed of value if you do what Torah says. But if you are a transgressor of Torah, your circumcision has become uncircumcision! 26 Therefore, if an uncircumcised man keeps the righteous requirements of the Torah, won't his uncircumcision be counted as circumcision? 27 Indeed, the man who is physically uncircumcised but obeys the Torah will stand as a judgment on you who have had a b'rit-milah and have Torah written out but violate it! 28 For the real Jew is not merely Jewish outwardly: true circumcision is not only external and physical. 29 On the contrary, the real Jew is one inwardly; and true circumcision is of the heart, spiritual not literal; so that his praise comes not from other people but from God." (CJB).
These verses reveal that Paul clearly understood the context of Exodus 12:48 to mean that YHWH accepts circumcision of the heart, meaning the man who takes YHWH as his God; that man is "of the circumcision." If not, if Exodus 12:48 only meant a physical circumcision was necessary for Passover, he would have said so!
That being said, we still believe that for a Goy/Gentile choosing YHWH as his God, the physical circumcision ultimately becomes necessary for it is and remains the "sign" given to Abraham (Genesis 17:11).
So concerning your question as to whether or not "Does that still hold true today [that a non-jew had to be circumcised to keep Passover]?", the answer is "no", it never was true. However, that answer carries with it a caveat. If the non-Jew has decided to obey YHWH, follow His Word, and become obedient to Torah, then that person becomes obligated to obey as many of the commandments as he can, and this includes physical circumcision. The physical circumcision, however, should not occur until "the heart is right."
Concerning the attendance of the uncircumcised at Passover:
Today at Passover, we marry the attributes of the Messiah's blood redemption and resurrection to life with the attributes of the original Passover, and our celebration is one of learning and growing closer to YHWH. If a physically uncircumcised man who has great love of YHWH as his God is in attendance and is learning and seeking to understand and preparing his heart for "the writing of the Torah" then - according to Paul (Romans 2:24-29 above) and a proper understanding of Exodus 12 - there is nothing wrong with his presence at the Passover meal. If this uncircumcised man is committing himself to letting go of his "old nature" and seeking the unleavened bread of sincerity and truth (1 Cor 5:7-8) then he is sure to reach that point, that epiphany, that wisdom, when he will realize that he can obey YHWH in the fullness YHWH expects if he submits, and he will want to become physically circumcised.
This man then, will have come to this point righteously, being filled with a deeper understanding, having grown closer to YHWH, rather than "just doing what Torah demands," or worse, doing that others say Torah demands. Afterwards, his NEXT Passover will be fuller, greater, and a more Ruach-filled experience.
By no means should any man become circumcised just to comply with Torah. Rav Shaul taught this clearly. Circumcision comes after Faith as it did for Avraham (Romans 4:12).
While many Rabbis may disagree with the idea of the uncircumcised being allowed to partake in the Passover meal, the fact is, the Apostles taught that one is to bring the Gentiles into the fold slowly, by not levying on them ALL the Torah at once, and helping them build and grow in their understanding, observing their growth, and allowing them to change their hearts such that when they reach that "ah-ha!" moment (i.e., when they "get it"), they no longer have to be taught by man, but by the Ruach haKodesh (Holy Spirit). Since the Messiah, the Living Word, was given to all, the participation in the Passover by a seeking, physically-uncircumcised Gentile is not only permitted, but highly desired.