Off the top of my head, the Roman Empire and the Republic used red prominently, as did the Soviet Union, as did Great Britain, and you can find red in the majority of coats of arms and flags. Why?
A few days ago, there was In almost all modern depictions of Roman legionnaire, he is shown in red. A red tunic, red-colored shield, or with red crest on the helmet. Is this accurate or is it modern convention? /u/Iguana_on_a_stick, whose flair here is "Roman Military Matters", replied that "the actual evidence is rather shaky" and, going by his conclusions, pretty well false.
As for heraldry: Wendy Erisman, "Trends in German Heraldic Style: An Analysis of Siebmacher's Wappenbuch von 1605", notes that a 51% majority of German coats from that source had red, but 68% had white.
I've just grabbed the most convenient source to hand: Cecil R. Humphery-Smith, Anglo-Norman Armory Two. It's an ordinary of arms of a number of thirteenth-century armorials. I flipped open to a page in the middle and counted tinctures on coats of arms for 20 pages: pp. 246-265 (chevrons between). I decided to count tinctures on distinct designs rather than holders, and I counted furs as their component colors (ermine = sable + argent).
118 designs.
The Wikipedia section "Frequency and national variants" is not so bad. I have the two sources footnoted as 3 (this Pastoureau is the little paperback from Abrams in 1997) and 39. It's Pastoureau that shows a unlabelled bar chart showing red well above white and yellow, but well below their sum (and his text contradicts the diagram: "a surprising emphasis on the colour blue" when the bar chart shows it as 5th of 6?!). Neubecker just says "in order of frequency of their appearance are red, blue, black, and green".
In my opinion, heraldry has few answers to the question "why?", except in a few obvious cases (like ermine being noticably more frequent in Brittany, where the count's coat was ermine). Heraldic treatises loved to talk about symbolism and meaning, and I think it's a load of bollocks Coglioni.
If I may be allowed to speculate: (1) there were only five common colors, so it's not improbable that something would be more popular, (2) red pops out to the viewer -- going back to a few pages of ANA2, red was the background on 9 coats but for charges (details, bits on the design) in 12, (3) red has the best contrast with the other heraldic colors, so in central Europe where you find color-on-color, one of the colors is usually red.
I believe it is because red works as a color for so many issues. As the color of blood and the expression of anger, it has always been a threatening and impressive color and it has always drawn humans to use it.
Red stands for blood. Many countries use red to represent their heroes of war and the blood they shed fighting for their fatherland.
Red stands for power. You already mentioned the Roman Empire, which started the trend among European medieval countries that red, more precisely Royal Purple, stood for rulership and royalty. Red thus became a color of power and the color you used to call back to the Roman Empire. For the reason of this, red to this day is prominent within the catholic church, as the color bishops and cardinals dress in.
Red stands for the working class. Starting with the French Revolution, red has recently been established as the color of the working and lower classes and its struggle for survival and rights. Hence the Soviet Union, which you also mentioned, picked red as its color. Red banners became a symbol of socialism and communism (comparable to how white banners stand for surrender and peace) and many short-lived Soviet or Socialist states used red flags as early solutions. Examples are the Hungarian and Bavarian Socialist Republics.
Red is a classic color of heraldry. You traditionally use the "metals", white (silver) and yellow (gold), with the "colors", the traditional ones being red, green, blue, black and purple. Many flags in Europe follow these rules, for example Austria, which put the metal silver between two red stripes. As it is frowned upon to put yellow next to white (this is reserved to the flag of the Vatican), you are supposed put metals next to colors and vice verca if at all possible. Not all countries do this. Germany for example has a flag from a time where the rules of heraldry were not that important anymore, thus they, while including yellow, put it under red and black, which border one another.
Source:
Fox-Davies, Arthur (1909). A Complete Guide to Heraldry (in the version of 2004)