I agree with some of the sentiment around the first point - certainly in a game centred around head bashing, not having the stats to bash heads in (the hypothetical all-3 statline in D&D, for example) can be annoying. I have played a few characters in similar positions though, and I will say that there is an interesting challenge in finding solutions without rolling when you know that you won't win a head-on conflict*.

I don't like point-buy systems though, as it tends to produce stat lines that are so very predictable - paladins always have 18 charisma + racial bonus (probably one that gives a +2 in charisma), so on and so fourth, and I see random rolls as the lesser evil. That's partially a matter of taste, I suppose.

Now, there is one rather nifty method I heard of a little while ago that gives randomised stats whilst also ensuring that all characters have an 'equal' statline. It also requires a symmetrical distribution (which the standard D&D 5th edition doesn't have - although as the 4d6 discard lowest is designed to avoid low-statline characters through sheer weight of probability, you could probably switch it to a 3d6 per attribute and then use this system) and it also requires an even number of attributes (though that can be worked around.)

What you do is you roll half of the attributes, then for each value take the equally probable value from the opposite side of the distribution curve. For 2d6, for example, the mean is 7. That means that if you roll a 7, a 9 and a 6 then the additional numbers you generate are 7 (from 7), 5 (from 9) and 8 (from 6.) Using this method, you are guaranteed to get an average of 7 but the results are still random - kind of nifty, really. Now, it will produce a more limited range of outcomes but that kind of shows it as it is - the half-way house between no randomness in generation and complete randomness in generation; I find that it's generally best used for games with more attributes, or that have a greater range of attributes.

As to maps, I think that stating that you can't do combat well without one is untrue. As a counterexample, the Skyward Steel naval combat expansion for Stars Without Number has really good ship-to-ship combat, with crew-members manning stations and choosing which actions they take to buff their ship or debuff their opponents. The system doesn't use a maps as they aren't considered necessary for the system - the focus is on the crew's choices on board. You could try and throw in Homeworld-style positioning combat, but that would add rather a lot more work. This extends to many other systems - as a rule of thumb combat should be as complex as it needs to be, but adding more can make the system worse by slowing it down without adding enough benefit. I agree that in a lot of cases, such as D&D, a map can add a lot - but that is system- and situation-dependent^ rather than an absolute statement.

To put it into a more abstract sense, I propose that one should add physical dimensions to combat when motion in that dimension has sufficient benefit (in terms of improved gameplay) for the cost involved (the cost being complexity.) Quite when that is varies by group.

For some situations, one dimension might be enough (a chase, for example, where the only real measure is distance from the target) whilst squad-based tactics appreciate two or even three. Whilst the idea of a four-dimensional combat, in which characters move through their timeline (say) and you can mess with cause and effect, sound really fun, I can't imagine how you would implement that in a practical manner on a tabletop (a good idea for a computer game, perhaps, because computers can handle a lot of complex systems without bothering the players with them.)

*probably my favourite was a 40k Only War campaign vs the Tau. Smart missiles, pulse rifles and railguns made combat a really nasty business for our squishy guardsmen, but a variety of tricks and clever ideas allowed us to achieve our mission regardless. 

^a good example is Legend of the Five Rings, which has a very interesting duelling system where a map is completely unnecessary, but it still has quite good depth to it (it's all about choosing when to strike.) 

That stat generation method guarantees you will have crappy numbers and good numbers. My hang up is in the idea that I could have had a better roll, theoretically. If the rules require that stat method then I guess I can't argue with the rules. But the end result is not much different from a point buy. You have the same total number of points in both cases, and in a point buy where you don't have enough points to have all max stats (else why even have points) the result is the same. In other words, if I spend enough points to get a 12 then it means I have to leave a stat at 2.

I played Legend Of The Five Rings one semester and I hated the game. Mostly because of the style and not the mechanics. I disagree that you can have good combat without a map. What you have is strategy, but not tactics. An abstract board game can have strategy. Rock paper scissors has strategy. But in a combat oriented rpg what I want is combat simulation.

Consider a minis war game. I want to place my units to utilize their abilities, which simulates real combat. I do NOT want to abstract that out to a point where we roll dice in a vacuum and declare that a battle had been won. I see no fun in merely looking at a page of numbers and buffing or cursing the numbers on a target's page. My ability X defeats your ability Y. I want to know that I won or lost because of my ability to manipulate the complex interacts of a physical environment and character ability. I want to actually flank a target, not play a card from a deck that says "Flank. Add 2 to your die roll." You could argue that a round of Magic The Gathering and a battle of Warhammer 40k both simulate a fight, but you can't say they satisfy the same desire of play.