'Random' can be totally fair. It depends what you do. For example, all the players could roll a set of ability scores, and then each player can choose whatever set they want, even if the same set is chosen multiple times. Guaranteed to be 'fair'.
Try this: get a normal deck of playing cards, take one red suit and one black suit, discard all aces, twos, threes, tens and pictures.
You now have two suits of six cards each, numbered 4 to 9.
I like randomness mixed with a bit of control. Too much control leads to cookie-cutter PCs and too little need for creative problem solving. Too little or no control leaves you helplessly tossed by the winds of fate.
So, bearing that in mind, take your twelve cards, shuffle them, and deal them all out face up, one at a time, to each stat, in order Str Dex Con Int Wis Cha. Then deal a second card to each.
Each stat will have two cards, ranging from four to nine. The worst a stat can be is to get both fours (8). The best would be both nines (18). The average is 13.
Right now, you have had no control over which stat got which score. Now, each player has the option of choosing two cards and switching them. You might choose to make sure you have an 18 (if you didn't already deal both nines to the same stat) but you only have a choice of two stats to make 18 because you only have a single switch so the 18 can only be one of the two stats that got dealt a nine.
You might have other priorities. Classes you are going for, party balance, all sorts. But you have a semi-random set of stats which has exactly the same total and stat range as every other player at the table.
You might think that this works fine when everyone is making a PC at the start of the campaign, but what about a replacement PC for when a PC dies? The randomness might mean that the vacated role cannot be adequate filled. Well, either suck it up think of a creative solution, or deal the first six cards out (one to each stat), and then arrange the remaining six cards as you like; there is no card switch in this version.
TL;DR: you can indeed have random scores which are nevertheless perfectly fair at the end, not just a perfectly fair chance but a bad result.