In other words, it was a deliberate attempt to improve the game by making a wider range of its play experiences accessible to its players.
Well, presumably all the 3e rules were designed to improve the game. The question for me is did they succeed, even by the stated goal you provide.
Q: Did they provide for a wider range of play experiences?
A: Not really. End game material had been published for earlier editions of D&D that suggested play at 20th or higher wasn't really necessary to encompass the full range of possible play. For example, all early adventure path end games - Queen of the Demonweb Pits, Dragons of Triumph, and Lost Tomb of Martek - end well before 20th level, and yet there is basically no feature of high level play not demonstrated in these modules. Mass combat, fights against deities, planar travel, epic scale, and large impact on the campaign world are all featured even though play is basically expected to top out at around 15th level at most. What exactly did we gain stretching play out for 5 more levels except for more grinding and filler added to a story line to ensure leveling took play?
Q: Did they make high level play more accessible?
A: No, not really. Arguably, older stand alone modules for high level characters provided much more accessibility if you wanted a high level fix. Simply play the module stand alone with constructed characters. Grinding down an adventure path still takes a considerable amount of commitment.
And consider the circular reasoning that is going on here. In 3e you need to be around 20th level to face epic foes like balrogs, pit fiends or ancient dragons. But in 3e epic foes enjoyed massive amounts of power bloat to ensure that 3e PC would have something to face at high level. So in making high level play accessible, they've pushed further away the tropes of high level foes and actually made that style of play less accessible. A very good example of rolling back that insanity is the 3e module, Lost City of Barakus where the designers carefully deflate the numbers on everything while still retaining a full and arguably epic campaign story complete with classic villains like lich kings. That made high level play more accessible. All 1-20 did was inflate the numbers without providing for more possibilities.
Q: Did they make high level play more enjoyable, more balanced, and more interesting.
A: Not at all! In fact, they made things much worse. All fortune mechanic systems tend to break down as the upper range of modifiers begins to meet or exceed the range of possibilities in the fortune mechanic. For D20, when the modifiers get to be around 20, the logic of the fortune mechanic tends to start failing. Things that are trivial for one character are nearly impossible for another. Spotlight sharing becomes more difficult and balanced challenges become harder to create. Plus, large numbers are always just more unwieldy to work with than smaller numbers (if you aren't on a computer), and the number of dice clattering that must be tabulated increases and slows down play. And when large numbers become required, they tend to be composed in practice of a bunch of fiddly small numbers that must be tracked and accumulated to be effective. Plus as the numbers on the character sheets get bigger, you increasingly risk a similar inflation in the number of actors in an encounter. So by making the levels higher, all your book keeping is getting more complicated, all your balance is getting harder, and the effort required to play is increased relative to the enjoyment received.
Q: Did they address what happened beyond the highest level of expected play well?
A: No, far from it. In 1e if you can assume that the normal highest level of expected play is roughly 10th-15th level based on the level by XP charts, the fullness of the game once you reach 10th level, and the level of published modules, then the published rules set is clearly sufficient in that if your cleric is 16th level you haven't hit a hard cap in the rules beyond which we can't peer, but merely a soft cap based on what is practical. But in 3e the rules break down completely after 20th level, with no satisfactory rules for what things are like after that ever published. Indeed, if we judge high level in 1e as 10th-15th, and look at the corresponding level of play in 3e as say 14th-20th, we find that in many ways they've actually narrowed the possibilities of play unless you are such an iron DM that you can dare deal with the problematic 'Epic' rules and all that implies. If you had played 1e up until 18th or 20th, you would have been doing the things you have to wait until 25th or 30th to do in 3e, and by that point "Good luck" because the math starts breaking down completely all over the place.
So regardless of what they were attempting, it had the appearance of not being particularly thought through.