I may be going out on a limb here, as this is only my opinion and not based on any researched facts, but I think in order to understand the revered reputation Randalls have, one needs to look at the era in which they gained that reputation. Keep in mind I'm primarily talking about fixed bladed knives here.
From the 1930s until the mid 1980s, there were very few commercial knife companies making good, strong, quality knives. Sure, there were custom makers, but not the plethora of them that there are now. Also, there was no internet then, and finding quality makers required going to knife or gun shows, or if you were lucky, finding a knife shop that carried some custom maker's knives. Randall's reputation was spread predominantly through "word of mouth." The quality of their hand-made knives was almost unheard of during those decades. I can't think of a single knife manufacturor during that time frame that comes even close to the quality of a Randall. Ruana's and Eks, though hand hand made and from the same (roughly) era, were not comparable to Randalls in their designs, overall fit and finish, or even in function (my opinion). I think Morseths came close (also my opinion). You didn't see many full-tang knives like the model 14 and 15 Randalls. The Buck model 124 comes to mind, but I don't think it was around until the 1970s. I believe the Puma White Hunter was/is full tang also, but off the top of my head, I can't think of any others, though I'm sure there are some.
By the 1980s, we start to see knife companies coming along that were making much better quality knives. Al Mar, Gerber (gasp!...prior to Fiskars buying them that is), Buck, Blackjack, SOG, Cold Steel (double gasp!) all come to mind when I think about quality knives in the late 1970s through the 1980s. We start to see full-tang knives with micarta slab scales becoming common for both custom and comercial makers in the 1990s. We see the Becker and RAT designs appear on the market. Awareness of custom makers becomes more predominant through the internet. There were many improvements in metallurgy as well over the last few decades.
I guess my point is that during the first three to four decades of Randall knives' existence, there were almost no knives that could compare to them. Nowadays, however, there are many knives that can, but that doesn't lessen the quality and usefulness of Randall knives and the reputation they earned.
BTW, I' basing my opinions on almost four decades of knife collecting and use. I own close to 500 knives, and have numerous examples of all the brands I mentioned. Of all my knives, my Randalls are my favorites.