I'd like to quote Rabbi Ari Storch, discussed at length here and here, which posits that, indeed, the witnesses were completely unnecessary for Kiddush HaChodesh, and were there only for ceremonial purposes. In these articles, he gets into the opinions of Rishonim, Geonim, and even Tannaim on the topic. For the purposes of this space, I will only focus on the main opinion, and the curious reader can view the links above for some fascinating applications of this principle.
Rabbeinu Bacheye cites from Rabbeinu Chananel that the highest beis din was charged with determining Rosh Chodesh, the new month. Rabbeinu Chananel states that those who maintain that it was necessary to have witnesses sight the new moon and testify before the court are mistaken. The court would determine the time of the new moon and based on their calculations sancitfy the months. The information contained in Maseches Rosh Hashana about witnesses is only because witnesses were used only for ceremonial purposes because of an event that had occurred at one point in time. However, the biblical commandment to sanctify the months has nothing to do with witnesses. (Rabbeinu Bacheye Shemos 12:2) In addition to not needing witnesses, the beis din is authorized to make Rosh Chodesh on a day other than the date of the new moon if it is necessary. Additionally, if they sanctified a day erroneously the sanctification is still valid. (Rosh Hashana 25a)
The above certainly is how Rabbeinu Bacheye cites Rabbeinu Chananel. It is of note that the Rambam emphatically rejected this opinion and felt that even Rabbeinu Chananel himself was not serious when he suggested it. (See Rambam Peirush Hamishnayos Rosh Hashana 2) Nevertheless, many Rishonim followed in the footsteps of that which Rabbeinu Chananel actually stated and there are indications that this was Rav Saadiah Gaon's opinion, as well. (See Torah Sheleimah Miluim Parshas Bo)
H/t to Isaac Moses for directing me to Rav Hirsch on Shemos 12:2. There he writes very similarly to this opinion:
From this mitzvah of קידוש החדש כל פי הראיה – i.e., determining the beginning of the month by the actual sighting of the recurring new light – those who never tire of disparaging the Jewish past in the eyes of the present generation have drawn an absurd conclusion: Originally, they say, the Jews lacked astronomical knowledge of the lunar cycle. Only centuries later did they learn from the Greeks the necessary methods of computation. Until that time, they had to content themselves with the crude makeshift practice of waiting each month for the reappearance of the moon.
We have called this argument an absurdity because it is obvious that no calendric system can be based solely on the sighting of the new moon. In a cloudy period, weeks can pass without proper visibility. Moreover, the very expectation of sighting the moon assumes a prior calculation of its probability. Indeed, from the תורה שבעל פה we learn that the determination of New Moons was based on calculating the moment in which the moon was to reappear. The results of this calculation served as a check on the testimony of the witnesses. What is more, if no witnesses appeared on the thirtieth of the month, the thirty-first became the first of the next month, even without the new moon having been sighted (see [Rosh Hashanah] 22ff.).
David could say to Yonasan: הנה חדש מחר [behold, the new moon is tomorrow] (Shemuel I, 20:5); thus, he knew beforehand that the morrow would be the first of the month. Moreover, from ויהי ממחרת החדש השני (ibid. 20:27) it is clear that, already in Sha'ul's time, they celebrated, as in our own time, two days of ראש חדש – an institution that is based on the calculation of the lunar cycle and the alternation of the months, מלא [thirty-day months] and חסר [twenty-nine-day months].
The word השני (ibid.) cannot mean the second day of the month, for the whole arrangement between them depended on the fact that, in honor of ראש חדש, the king would hold a banquet at which all the members of his court were expected to appear. Clearly, then, השני means the second day of the New Moon, as Targum Yonasan translates it: והוה ביומא דבתרוהי דהוא עבור ירחא תנינא.
I'm actually struggling to find a Rishon who holds as you say, that the calculations are irrelevant. The Rambam writes (Hil. Kiddush HaChodesh 1:6) that the calculations are there to determine the validity of witnesses:
בֵּית דִּין מְחַשְּׁבִין בְּחֶשְׁבּוֹנוֹת כְּדֶרֶךְ שֶׁמְּחַשְּׁבִים הָאִיצְטַגְנִינִים שֶׁיּוֹדְעִין מְקוֹמוֹת הַכּוֹכָבִים וּמַהֲלָכָם וְחוֹקְרִים וּמְדַקְדְּקִים עַד שֶׁיֵּדְעוּ אִם אֶפְשָׁר שֶׁיֵּרָאֶה הַיָּרֵחַ בִּזְמַנּוֹ שֶׁהוּא לֵיל שְׁלֹשִׁים אוֹ אִי אֶפְשָׁר. אִם יָדְעוּ שֶׁאֶפְשָׁר שֶׁיֵּרָאֶה יוֹשְׁבִין וּמְצַפִּין לָעֵדִים כָּל הַיּוֹם כֻּלּוֹ שֶׁהוּא יוֹם שְׁלֹשִׁים. אִם בָּאוּ עֵדִים וּדְרָשׁוּם וַחֲקָרוּם כַּהֲלָכָה וְנֶאֶמְנוּ דִּבְרֵיהֶם מְקַדְּשִׁין אוֹתוֹ. וְאִם לֹא נִרְאָה וְלֹא בָּאוּ עֵדִים מַשְׁלִימִין שְׁלֹשִׁים וְיִהְיֶה חֹדֶשׁ מְעֵבָּר. וְאִם יָדְעוּ בְּחֶשְׁבּוֹן שֶׁאִי אֶפְשָׁר שֶׁיֵּרָאֶה אֵין יוֹשְׁבִים יוֹם שְׁלֹשִׁים וְאֵין מְצַפִּין לָעֵדִים. וְאִם בָּאוּ עֵדִים יוֹדְעִין בְּוַדַּאי שֶׁהֵן עֵדֵי שֶׁקֶר אוֹ שֶׁנִּרְאֵית לָהֶם דְּמוּת לְבָנָה מִן הֶעָבִים וְאֵינָהּ הַלְּבָנָה הַוַּדָּאִית:
The court calculates, in the manner which astrologers calculate that they know the places of the stars and their paths, and they probe until they know if it's possible that the moon can be seen in its time, which is the thirtieth night, or if it's impossible. If they know that it's possible that it can be seen, they sit and wait for witnesses the entire thirtieth day. If witnesses come, and they probe them according to Halacha and their words are confirmed, we sanctify it. If it's not seen and witnesses do not come, they complete thirty and the month is large. But if they know from the calculations that it cannot be seen, they don't sit and wait for witnesses the entire thirtieth day, and if witnesses come, we know for certain that they're lying, or that they saw clouds which looked to them like the moon, but it was not certainly the moon.
The Ramban in his comments to Sefer HaMitzvos, Asei 153:
ומה שאמרו (ב"ב קכ"א) מועדי ה' צריכין מומחין ומועדי ה' צריכין קדוש ב"ד זהו למצוה, א"נ צריכין ב"ד קאמר לחשוב בהן ולהסכים בהן אם מלאים אם חסרים ובשנים אם פשוטה או מעוברת וכן אמרו (ר"ה ז') שתא מעוברת בחושבנא תליא מילתא
And that which they said (Bava Basra 121a), that the "holidays of Hashem" need experts, and the "holidays of Hashem" need sanctification of Beis Din, this is only for the mitzvah, or "need Beis Din" means to calculate and agree with them if they are full or deficient, or by years if they're regular or leap years. So did they say (Rosh Hashanah 7a), "Leap years are dependent on the calculation."
The Sefer HaChinuch (§4):
מצות קידוש החודש: לקדש חדשים ולעבר שנים בבית דין גדול בחכמה סמוך בארץ, ולקבוע מועדי השנה על פי אותו קידוש, שנאמר, החדש הזה לכם ראש חדשים, כלומר, כשתראו חדושה של לבנה תקבעו לכם ראש חדש, או אפילו לא תראוה מכיון שהיא ראויה להראות על פי החשבון המקובל
The Mitzvah to sanctify the month: To sanctify the months and intercalate years in the great ordained court in the land, and to fix the holidays of the year according to that sanctification, as it says, "This month will be for you the head of months," that is to say, when you see the birth of the moon, fix that for yourselves as the head of the month, or even if you don't see it, since it is fitting to be "seen" according to the received calculations.
Nobody seems to say that it absolutely must be done by calculations; everyone seems to hold at a minimum that the witnesses are ideal but not required.