It was a way of showing that Adoniyahu was trying to take over the kingship improperly. Natan was the navi who would have stopped the attempt by declaring what Hashem wanted and Shlomo was the actual heir whose position was being usurped. This showed that Adoniyahu knew that he was not supposed to be the heir even though he was now the eldest.
Melachim I 1:5
וַאֲדֹנִיָּ֧ה בֶן־חַגִּ֛ית מִתְנַשֵּׂ֥א לֵאמֹ֖ר אֲנִ֣י אֶמְלֹ֑ךְ
וַיַּ֣עַשׂ ל֗וֹ רֶכֶב וּפָ֣רָשִׁ֔ים וַחֲמִשִּׁ֥ים אִ֖ישׁ רָצִ֥ים
And Adoniahu the son of Haggith exalted himself saying: 'I will be
king,' and he made for himself chariots and horsemen and fifty men to
run before him.
From the word אֶמְלֹ֑ךְ we see that he learned from Avshalom and carefully did not attempt to overthrow his father. Instead in arranged for everyone to assume that he was the heir as the oldest surviving son of King David.
And his father had not angered him all his days saying, "Why have you
done so?" And he too was of very handsome appearance, and she bore him
When he invited all of the major people in the kingdom, he was very careful to do it in such a way that they would all assume that he was indeed the legitimate heir. This would mean that when King David died, he would get the crown as it would be too late for Shlomo to raise a fuss.
That is why he very carefully did not notify the three people, who had they been there would have stopped him. Ralbag explicitly says that they would have objected and stopped him had they been there.
And Nathan the prophet and Benaiahu and the mighty men and his
brother, Solomon, he did not call.
And his brother Solomon, he did not call: for he knew that the prophet
had prophesied about him that he would reign [after David’s death].
That is why King David immediately had Shlomo anointed as the next king 1:32-40 And Adoniyahu immediately gave up his attempt and acknowledged Shlomo as king.