Tens of thousands
While I agree with Nobody's conclusion that, restricted to the constraints proposed, this is not a fight between an 'ancient red dragon' and 'commoners' in any meaningful sense. However, I don't think it is 'useless other than as a math diversion', because it illuminates some design features of the combat system. Making an approximate comparison can illustrate things such as the limited ability of CR to understand relative strength in combat. Things like movement speed and having the ability to fly are essential to understanding tactics and likely outcomes, and yet they have no effect on CR. An exercise like this points out some crucial assumptions and differences between editions. In first edition, once you reached a certain AC, you became unhittable since a 20N was not automatically a hit. The fact that in 5e a 20N always hits (but does not necessarily save) means that 'an army of commoners' is a more credible threat, and recognizing that things like proficiency bonus and ability modifiers don't matter in this fight shows how the 20N rule presents its own challenges to bounded accuracy. The exercise also illustrates how the 20N rule incentivizes arming your army of commoners with ranged weapons. To strain a metaphor, historically, once the damage capability of ranged weapons outpaced the protective power of armor, so that high-AC individuals were no longer unhittable, battlefield superiority shifted from high CR individuals (mounted knights) to an army of commoners with ranged weapons (longbowmen and eventually musketeers).
Back of the envelope calculations
Assume an infinite plane with an infinite number of commoners, regularly spaced at one per each 5 foot hex.
Big simplifying assumption - The dragon controls 19 hexes. In reality, when playing on a grid, a huge creature is supposed to control 7 hexes and a gargantuan one 12 (DMG 249). The huge creature is a central hex surrounded by a single ring of six hexes, but the 'footprint' of a gargantuan creature is not an easy shape to describe. I made the dragon a single hex surrounded by two rings of hexes (6 and 12, for a total of 19) in order to visualize everything as concentric rings of hexes, which kept the math far easier.
Assume the dragon kills about 144 commoners each round. Its Wing Attack will automatically kill any commoners in its area of effect (they can't make the DC25 save, and even the minimum damage is more than their hp). If the dragon is 'three rings' of hexes, then everyone within 15' is the commoners in rings four (18), five (24), and six (36), for a total of 78 commoners in the area of effect. This is its best attack. Add to that its breath weapon - all commoners in the area automatically die, as above. A 90' cone (affects 1 hex in the first rank + 2 + 3...19) will kill 190 commoners, but breathing every third round means an average of 63 per round. We can add to that two-thirds of a bite, four-thirds of a claw, and (depending on your assumptions about initiative) two-thirds of a tail. OP would like the dragon to miss on a 1N, but the numbers it can kill with the melee attacks are so small compared to the area effects that we needn't bother. We'll round it to 3 and say that 78 + 63 + 3 = about 144 commoners killed per round.
Any commoner within 60 feet can attack without moving - A commoner can throw its club up to 60 feet as an improvised weapon. How many commoners are within 60 feet? Keeping within our concentric rings, the first, second, and third rings of hexes are the dragon itself. Ring four is adjacent to the dragon; 18 commoners can attack the dragon in melee. Ring five is 5' from the dragon; rings five, six (10'), seven (15'), and eight (20') can make short-ranged attacks, for a total of 132 thrown clubs. Rings nine (25') through twenty (60') can also make ranged attacks, but at disadvantage from long range, so that's another 962 clubs thrown with disadvantage.
All told, that's (18 + 132 + 962) = 1112 commoners who can attack the dragon, each round, without even moving. Of course, the dragon will be 'opening a space' for itself as it kills commoners. Given that it will be killing only 144 of these 1112 commoners, evenly distributed that is all the commoners in rings five through eight plus a few more, and that is only twenty feet from the dragon. Since the commoners have a move of 30', they can fill in the gaps made by the dragon without losing any attacks, and thus they can fill any space faster than it can kill them.
Consider, however, OP's constraint that the dragon cannot use its Frightful Presence. This is key to even running this analysis. The range of this feature is 120', meaning that by taking its action every turn the dragon could keep all commoners at 120 feet or further (they cannot make a DC21 Wisdom save). Since the commoners have to start their turn at 90 feet or closer to attack (30 feet move plus a 60 foot ranged thrown club), simply by re-using this ability the dragon can hold off the commoners indefinitely. A canny dragon, since it can choose who the presence affects, could alternate between frightful presence and wing attacks to kill commoners indefinitely.
FIRST CONCLUSION: Without frightful presence, since the dragon cannot kill the commoners as fast as they can approach it, it cannot hold them off indefinitely. It will eventually be killed; it is just a question of how long it can continue to kill commoners, and how many it will take out before it dies.
Knowing this, we can set some rough bounds on how long this might take. On the low end, suppose the dragon was subjected to the aforementioned 1112 attacks each round. With its AC22, the dragon is only going to be hit on a 20N, which is convenient. The 150 melee and close-range attacks have an expected damage of (1/20)(5)(150) = 37.5 per round, while the 962 long-range attacks with disadvantage have an expected damage of (1/400)(5)(962) = 12.025 per round, for a total of 49.525. With 546 hp, we can expect our dragon to go about 11 rounds against a plane filled with commoners.
At the high end, suppose there is an empty space in a circle with a radius 75 feet across and the dragon in the center. Only the outermost (20th) ring and beyond is filled with commoners, such that the dragon is receiving only 114 attacks, all of thrown clubs at disadvantage, and the commoners are not approaching any closer. In that case the expected damage is (1/400)(5)(114) = 1.425 per round, and the combat will last about 383 rounds.
If we ran this as a tactical battle on a map, with the dragon attempting to use its movement and area attacks to continually open a space and then stay as far from the commoner attacks as it could, and the commoners continually attempting to move closer to the dragon and attack, the results would have to be somewhere between our two bounds, somewhere between a plane with every space filled by commoners and one in which all commoners remained at 60 feet distant from the dragon. Thus, we can expect an actual tactical combat to last somewhere between 11 and 383 rounds. Given that our dragon will kill 144 commoners each round of combat, we can expect it to have killed between c. 1600 and 55,000 commoners by the time it goes down.
Note that this is not exactly OP's question, which is 'what is the lowest number of commoners that could defeat an ancient red dragon'. Instead, it assumes an infinite supply of commoners and asks how many will the dragon kill before it goes down. If the commoners were, say, an army of 50,000, they would be deployed finitely in space. In this case the dragon could, as Thomas Markov says, simply keep ahead of any advancing formation using its superior movement, only allowing them to close the distance occasionally so that it could breathe on them. This strategy would work until they were so many that they could effectively surround the dragon at any scale and then close the distance to it, with it unable to escape without cutting through them to get to the 'other side'.
SECOND CONCLUSION: Without being able to fly, the number of commoners required to defeat the dragon would be the number required to deploy in such a way so that they could surround it with no chance of escape - which gets into questions of their deployment and the dragon's advance knowledge that OP does not specify. Once the commoners had the dragon surrounded, they could move in until they were in attack range, at which point several thousand to tens of thousands of would die before they were able to bring the dragon down.
