So your question is addressed by Rav Yosef Chaim Sonnenfeld zt"l in his Shu"t Shalmas Chaim 326.
He begins by asking why one would not dip the challah in salt on Rosh Hashanah and he answers initially:
לכאורה יש לומר, אחרי דנוהגין לטבול פרוסת המוציא בדבש, ואם יתן עליה גם מלח יהיה נותן טעם לפגם, ואין ראוי לעשות זה בליל ראש השנה, והרי העיקר שהוא טובל בדבש הוא לסימן טוב על שהוא מתוק
At first glance there is to say, that there are those who are accustomed to dip the slice of hamotzi in honey, and if you also place it in salt it will impart a flavour that ruins it, and this is not fitting to be done on the night of Rosh Hashanah when the main thing is to dip it in honey as a siman - a good sign that it (i.e. the forthcoming year) should be sweet.
He reasons therefore that we should avoid using the salt when we have the honey. He brings the Be'er Heitev 167 who expands on the Rema's words and gives more Kabbalistic reasoning for the salt, however...
מכל מקום כאן אי אפשר
Nevertheless, here it is impossible
And as far as the Rema's point about the table being compared to a mizbeach (altar) and therefore requires salt, he writes that the implication is:
דהבאת מלח על השלחן אפילו אם לא יטבול הוא בעצמו מצוה
Bringing salt on the table even if he does not dip is itself a mitzvah.
Accordingly, even if we don't dip our challah in salt, we still make a point of bringing it to the table.
However, he counters as per the fourth perek of Tamid there was salt required afterwards on the meat of the Korban, and if so, the salt would need to be on the food (which is compared to the Korban) and it is not enough for it to be just on the table.
Interestingly, he therefore takes the middle ground and concludes:
תשובה: בפשיטות שוודאי נכון להביא מלח ולטבול בדבש ולטעום, ושוב אני טובל במלח, אבל לא בפעם אחת כי אם משני צדדים, ואני טועם מקודם הדבש עם הלחם לסימנא טבא, ושוב אני טובל במלח ג"כ ולהזכיר ברית מלח עולם
Answer: It is definitely correct to bring salt, and dip in honey and taste, and I (make a point of) dipping again in salt - but not once because of the two sides. I dip first in honey with the challah as that is as good sign (for the forthcoming year) and then I dip again in salt also to remember the everlasting covenant of salt.