We have been down this path about 30 pages ago. The answer is that it's not just about presentation. Now the other side will tell me that can't be the case but how can they know what it is or isn't about for me.
I'm not so sure it's just about presentation, either – but I think the presentation is a big part of how we got to this impression. It puts a lot of pressure on both the designers to create powers for every different aspect of play, and on players to choose the right powers when they level up (or from their spellbook, I getcha Essentials Wizards), and again during combat.
Other powers that anyone can use exist – they are skillful tricks that you might attempt with the basic rules and resolution of skills and weaponry and whatnot. But because those resolve at the whims of the DM, there is very little incentive in the Player-First 4e game to use them. You've got these shiny powers that only work once an encounter, and you're going to waste your round attempting to do some stunt that may or may not be effective?
5e still have at-will, encounter, and daily resources, and they have them in various classes, including the Fighter (Battle Master). I think the reason why some people see Battle Masters as the Instant Coffee as compared to Fighters/Rangers/Warlords in 4e is because their options and usage #s and the scaling of those options are very limited.
5e is purposefully designing to limit those, so that Martial Characters don't feel locked between what they can do on paper and what they can do in their heads. 5e Fighters get more ASIs than other classes for this very reason – with stat boosts, either their skills and attacks and defenses get swifter increases and they can perform more daring and mighty feats, or they get cool new feat abilities that greatly increase their arsenals (or lightly increase them, while also raising their stats). It's NEVER going to be as many abilities as in the 4e arsenal, because 5e is designed to to be less work on the designers (even for spellcasters – they're reusing spells and scaling them (a bit like 4e Psionics) rather than creating 5+ different ways of saying Fireball.
In that sense, 4e Powers are same-y because they reinvent the wheel for different magical classes rather than just giving them all fireball and making the spell malleable in its description to fit the different purposes.
On the other hand, you could say that 5e makes each of those different characters who get Fireball same-y, because they're all using Fireball, rather than their own unique iterations on the fire attack – from a primal perspective, from a divine perspective, from a psionics perspective, from a sorcerous perspective, etc.
I think one the trap there is that if your concept doesn't fit into the 4e matrix SOMEWHERE, you're either at a loss, or you have to refluff. So there's a huge onus on designers to make something for everyone there, but inevitably, as this is a very creative and personal hobby, someone is going to feel left out.* In 5e, they embrace the refluff potential, and ask you to stretch that limit and think about how different spells and actions and abilities could work differently on various characters. This eases up on the designers (there is far less pressure to churn out additional splatbooks for more essential character concepts), and it also allows for MORE character concepts to see the limelight because there's less work to make those concepts shine within the edition.
But if your character and ideas fit within the 4e matrix, and you can build the abilities YOU want to use** within a 4e character, then the game is very well designed for you, and this need not be a problem.
*Aside: Certainly Bard and Druid and Gnome and Half-orc etc Players felt left out in 2008. As a lover of Bards and Druids, I too was bummed, but I then decided to roll up an Eladrin Control Wizard and had a BLAST with a character I might never have made otherwise. And when PH2 came out, I had fun with my older characters finally making the jump to 4e.
**Second Aside: Strangely enough, I had an issue where I didn't WANT more powers at a certain level, I wanted to use my old abilities, but have them scale with me, or reflect my character's growth. 4e didn't ALWAYS have a powered-up version of the previous abilities you had, and I felt myself having to either sacrifice story abilities, power level, or the rules as written. I might not WANT additional powers, and I'm just getting them because that's what I get at this level. Or I might be at the level where I need to lose previous powers and replace them with new ones, in which case I might like my old Warden form and not want this shiny new one that doesn't fit my character. In that case, I may try to reflavour the new form, change up the damage type or condition rider to fit the old one, etc. But that's getting into a form of tinkering that can have real ramifications on the balance of the game, because not all damage types or conditions are equal, and that means getting the DM involved in MY character.
Then, of course, there's the CharOps forum that's telling me my character's abilities are purple or even the dreaded red, when I should only be using gold or sky blue options at that level. Too bad! That flavorful ability won't let me keep up with the rest of the party, and because we don't have Bounded Accuracy and so I REALLY have to make my ability and power investments matter at every choice point.
If I sound a little bitter, it's because I adore 4e AND 5e so much, and in my ideal D&D world, I'd have the flexibility and ability to scale my non-spellcaster powers to herculean abilities if I so chose, or the ability to dial it back toward something more like 5e's non-spellcasters, and still remained balanced as a team against the common threat. I think 4Essentials and then 5e TRIED to do this but neither implemented it as well as I'd like to see in a hypothetical 6e.
