HIGH-VARIANCE ROLEPLAYING SYSTEMS?

Most systems that use a flat distribution instead of a bell curve also don't start from zero with the dice. Instead of 3d6, you roll 1d6 + 6. (The latter number is probably based on the character's stats 99% of the time.) This actually makes the result less random, though, not more, so it doesn't really address your question.

There are times when I've used extremely high variance on purpose to address other problems.

For example, the default random variance for damage in most versions of RPG Maker is 15% or 20%. If your average enemy dies in only 1 or 2 hits, however, you may want damage for all your player's attacks to randomly vary by up to as much as 70% or 80%. If your players die in only 1 or 2 hits, you may want to do the same thing for enemy attacks.

What this does is make every point of attack power and defense power and HP actually matter. Without the high randomness, you run into a problem. If attacks are relatively consistent in damage, and the player can already kill the enemy in 3 hits... how many points of attack power does it take them to get to the point where it only takes 2 hits? If it takes 4 points of attack power, then any smaller increase in attack power is worthless. However, with enough randomness, every point of attack power is meaningful because it gives players a chance to win sooner.

This can be worth considering even if your game isn't really very heavily stats-based, since once this problem pops up, the meta-game for stats will become a big deal whether you want it to or not. If you solve it by introducing extra randomness, the players can stop thinking about their stats and just pick the obvious "best" piece of equipment.

Enemies in most game systems naturally vary in power compared to each-other, though, so if you don't expect players to heal between battles, then you should strongly consider making enemy attacks fairly predictable while making player attacks more random. It doesn't feel good to anyone when an enemy gets a critical hit and kills the player in what had a 98% chance to be a safe situation. No matter what your system is, you pretty much always want every outcome other than the "expected" outcome to be better for the player than what they expect. You almost never want to have a below-average random chance to favor the enemy. It just feels shitty. The one exception is when it's a random chance to face a harder encounter - this feels better than a random chance to fare worse during the same encounter, because the player also gets an extra sense of accomplishment (and extra exp and loot) for winning. Choosing which encounter the player should face doesn't contradict the player's expectations so much as it creates their expectations.

Tabletop games have high variability because you have a group of people who can echo off of each other and celebrate or lament/laugh off the results. A good GM can make rolling a 1 entertaining, even if it's not beneficial. That's not really the case with a computerized system. It's normally just you playing, and like LockeZ said, the game can sometimes feel like it's pulling one over on you. (I'd say "suck it up" as long as the rules of the game are transparent to the player, but whatever.)

Tabletop rules are meant to be broken and enjoyed as a group, and it just doesn't work as well on a computer because in digital RPGs, you can't break the rules. There's no overseer fudging the dice to enhance the narrative (usually). (The closest thing I've seen would be Hand of Fate, where having a GM to play against is the main conceit.)

I dunno if this is helpful at all, but at least this is why I believe tabletop rules don't translate well to digital rpgs.

@Aubrey: To answer your last question, you can have a look at Roll20. I've never used it but I watch streams and it seems quite user-friendly.
I suppose you could use the character sheets and dice rolls along with a real tabletop game.