I'm not convinced that RNG is required for an RPG.
Consider action RPGs where HP tend to be depicted more often as hearts. I believe in general you find that an enemy does a consistent amount of damage with each hit and your weapon will tend to be consistent as well.
And I know it's not an RPG but playing numerous hours of Civ4 I noticed that it's pretty rare that battles were decided on randomness. More often than not battles are decided on any terrain & unit tech bonuses and the number of units involved. In general, there was a logic to the outcomes of the battles and "fun" was in your ability to prepare for what's coming.
Which, in case of an RPG, can lead to boring battles getting even more boring.
If battles are already boring then shift away from their prevalence and focus more on preparing the player and his character(s) for other challenges or providing other elements of gameplay where the player can have fun.
Well, I am not saying "An RPG without RNG is boring!"... but just what I perceive as to why many games today choose to include some form of RNG somewhere (which can come in the form of a Physics engine, which in the end is just a fancy RNG that should LOOK like it models physical behaviour, for example)...
Now, I am in no way saying "Chess is boring", or "Game X is boring"... I am saying that missing an element of randomness improves the ability of the player to number crunch the outcome of battles. Which, if we are talking round based RPGs, leads to boring battles (as the only randomness comes now from the abilities the player, and maybe, depending on how good the AI is, the AI picks).
In Action Adventures, like Zelda for example, there is way more randomness included as well as a skill element, as the player now has to move around during battle, can evade getting hit this way or also make a mistake. An additional RNG is much less needed now and, given the "skill" aspect of the game should take the front seat, might actually be detrimental to the expierience, as the RNG might screw the player even though he/she plays very well, or might reduce the amount of skill needed as a lucky RNG result might make up for a mistake of the player.
I guess the Problem here is, what kind of RPGs are we talking about? Old JRPG systems with turn based static combat? Modern western RPGs which are more like FPS with RPG Elements and a Hack'n'slash system? Tactical round based RPGs? Action Adventures?
Generally, in single player games RNGs are used to spice up the system, and to spark the gambler gene in the player. That doesn't make it a good fit for all games, and certainly games can work without it.
Many RPGs though, especially the older ones, have pretty "simplistic" systems. There is a reason why many JRPGs had "auto battle" buttons so you didn't had to mash the attack button 20 times for every damned random encounter, which happened to pop up once every minute. These become pretty annoying when you are overleveled and these are just timesinks, not giving back sensible amount of exp or ingame currency, nor providing any challenge or novelty. The RNG alone will not change much in this situation... but it does spice up the battles that are more challenging. Which could also become pretty boring once you figured out the system. Have a skill that lets you analyze the enemies health, in a game without RNG? Great, now you can number crunch exactly how long you have to attack to bring the enemy down. All you need now is the attack pattern (which, completly without RNG, will also be extremly predictable), and you can pretty much write a bot that beats the enemy 100% of times, without any conditional logic or sensors needed. Just mashing the buttons in the right sequence.
There are also the odd bunch were the RNG actually worked against the game, because it had too much influence. I recently digged up my old SNES Breath of Fire II cartridge. I had completly forgotten how unbalanced the RNG was in this game. A single dodge or critical hit could really mess up your day or turn a close fight into an assured victory. High damage enemies, very high crit chance for both enemies and your own character and a lot of dodges really make every random encounter something of a gamble...
I think it is positive, mindless attack button mashing doesn't work all the time anymore, but on the other hand I had to redo one or two battles because the RNG screwed me over hard. It certainly made a difference to how you play compared to other games.
Personally, I'd say RNG is a good idea if a) you want to spice up a simple system to make it more interesting, b) you have a simple system that should model reality (in reality, everything is one huge RNG unless you understand how everything works down to the atom), or c) you want to reduce the skill factor (RNG usually levels the skill needed, as good player might be held back by the RNG).
Its a bad idea when a) you have a very complex system which simulates reality (though there are always things you cannot simulate, where an additional RNG might make sense to make things "more realistic"), b) you have a system that SHOULD be predictable and number crunching should be part of the game, or c) you want to make sure skill will be the main influence on the result of a player.