score:2
It is a good question. To resolve this, I identify the following issues and relevant points:
Whether there is a difference in the underlying Greek text edition used by KJV (Textus Receptus) vs. NRSVCE (Nestle-Aland).
Answer: NO. So it is not the case that KJV is more faithful to the original text.
Whether the ambiguity is due to the Greek text itself.
Answer: YES. The underlying Greek text left it as "they" (see BibleHub interlinear Greek for Mark 3:21).
Whether the translation difference you see in NRSVCE vs. KJV is about Catholic vs. Protestant.
Answer: NO, because as you can see in a list of English Translations the translations that use "people" are mostly RSV (denomination-neutral) based and a few newer dynamic Protestant translations (ERV, GNT, ICB, MOUNCE, and PHILLIPS).
Whether in the history of Mark 3:21 interpretation, Catholic interpretation of the referent of "they" differs from Protestants.
Answer: PROBABLY, as they tend to protect the implication that the Blessed Virgin Mary thought Jesus was "out of his mind" (see Dave Armstrong's 2020 article Did the Blessed Virgin Mary think Jesus Was Nuts? which points out the ambiguity mentioned above, citing the 1953 Catholic Commentary edited by Dom Bernard Orchard as well as recent dynamic translations).
How did the early church interpret Mark 3:21?
Answer: According to a journal article by Methodist Drew S. Holland The Meaning of Ἐξέστη in Mark 3:21 from the 2017 Issue 1 of The Journal of Inductive Biblical Studies) pages 27-29, it turns out that there was an East and West differences in the interpretation, with the West tended to be guided by Jerome (Jesus was out of his mind) while the East tended to be guided by Pseudo-Chrysostom (the crowd was amazed) whose view is preserved in Aquinas's Catena Aurea: Gospel of Mark:
Ungrateful indeed were the multitudes of princes, whom their pride hinders from knowledge, but the grateful multitude of the people came to Jesus.
CONCLUSION: Although both KJV and RSV strives to be literal, the RSV translator apparently tried to resolve the ambiguity inherent in the unspecified referent of "they" in the original Greek Text. Recent Catholic scholars (such as the 1953 Dom Bernard Orchard commentary), recent denomination-agnostic Biblical scholars (such as the 2017 paper by Drew S. Holland mentioned above), and recent dynamic translations (such as ERV, GNT, ICB, MOUNCE, and PHILLIPS) started to interpret the "they" as referring to "the crowd". Even the term Ἐξέστη itself is ambiguous, not necessarily meaning "out of His mind".
The aforementioned scholarly article has the following abstract (emphasis mine):
In examining Mark 3:21, scholars over the last century have focused their attention on the identity of οἱ παρ’ αὐτοῦ. The consequence is that scholarship has reached an impasse in determining who claims that Jesus has gone mad (ἐξέστη). The following paper attempts to focus instead on the meaning of ἐξέστη in Mark 3:21 as a key to solving the interpretational difficulties that have surrounded this verse and the pericope in which it is found (Mark 3:20-30). I propose that ἐξέστη means “he has amazed” as opposed to the traditional sense of “he has gone mad.” Moreover, it is the crowd, not οἱ παρ’ αὐτοῦ, who makes this claim about Jesus. This eases the exigency of locating the identity of οἱ παρ’ αὐτοῦ since we are no longer required to explain why either of these groups would claim Jesus’s insanity. This approach is strengthened by a literary pattern spanning Mark’s Gospel from the beginning until the passion narrative in which the crowd responds positively to Jesus, especially in contrast to religious leaders.
and thus makes the case to translate Mark 3:21 as:
“And having heard, the ones near him [the disciples or his family] went out to take hold of him; for they [the crowd] were saying that he has amazed [us].”