We shouldn't have to constantly refer to any table, that is why we should have a formula for modifiers.
The 1d20 with only a maximum modifier of +5 means that randomness means too much.
I'm saying that a maximum roll on anything should be 1d20+10. The +10 comes from ability30(ability modifier +10)
That bonus of +10 includes racial bonus, class bonus, level-up, magic items, spell effects and feat effects.
Take that ability modifier and apply it to skills, toHit, and damage.
Let the dexterity modifier and better armor improve the AC.
Capping at 20 is too low when you can roll an 18 for an ability. Capping at 30 will allow you to stack feats, items, effects and level-up into the ability.
For those of you who say that no ability should improve when leveling-up, I think you're in the minority. You're all agreeing with each other here but others don't.
If you still want a cap at 20, tell me how you would have leveling-up and magic items stack up in a bonus.
You seem to have a very particular and fixed vision of how characters will advance as they level up. We have *no idea* right now what will increase and when, and we have no idea how difficult high level hazards and monsters will be to overcome and hit respectively.
I will address your comment about 'randomness' though. A +5 modifier cannot be compared to the variance of the die roll in any meaningful way. You can only compare the *spread* of modifiers across several characters/levels/situations to the variance of the die roll. For instance, if the largest difference in attack roll modifiers we see is 5, this is less than the standard deviation of a d20 (5.77). In practice, it would feel like the better character wasn't that special, because the die roll is swingier than the difference between the characters. If the largest difference in modifiers we saw was +15, then it would feel like the better character was genuinely superior than the worse.
There is a tricky balance to be found between making the differences between characters seem meaningful, without producing situations in which a party cannot function in parallel. At high level in 3E there were huge differences between characters' save, skill and attack bonuses such that a DM would find it difficult to create an encounter in which everyone had a fair chance. Now I don't mind this with different skills, so different characters can shine, but it's not good when only some characters can hit or resist an effect, but others can't. 4E realised this, and took action to reduce differences across different characters as best it could. There were still fairly big differences in skills (acceptable), defenses (due to feats, less acceptable), but almost no difference in attacks (I don't remember a larger difference within a party of more than 2), which was maybe overcompensating for 3E.
The differences in 5E are on the same scale as differences in ability modifiers (-1 to +5, nominally), with another +-2 on attacks from class-specific features, +-2 on AC from shields (assuming they fix armour a bit) and +-3 on skills. Provided (god I hope) they don't introduce feats or abilities that allow any greater differences, we should be in good shape for functional, but diverse characters. They even have some design space to modify specific saving throws. I hope that the flatter math will keep differences <10!