I think part of the problem lies in the use of the word "magical" itself. Its definition can be very personal, which leads to us talking past each other. So I'm going to fork the question into component parts. Instead of using "feel magical," I'm going to use "invoke awe or appear mysterious." This is more specific, and thus clearer as to what I'm actually saying. "Magic" breaks into its now four prime elements:
cantrips,
spellcasting,
rituals, and
items.
Do cantrips invoke awe or appear mysterious?
No--except in unusual situations. Sometimes, such usable-on-command magic can be exciting or awe-inspiring (I'm thinking stuff like the 4e Wizard's "Cantrip" feature--not at-will attacks, but the unique non-combat spells like Mage Hand, Prestidigitation, etc.), but the vast majority of such magic is
intentionally workhorse. This does not have to devalue magic in your game...IF you can get on board with the idea that something *we* would consider flabbergasting is an everyday action to certain characters. It's the equivalent of transporters and replicators in Star Trek--to us, these things are incredible, but to a 24th-century Starfleet officer they're standard-issue. We feel like their world is MORE wonderful because something clearly wonderful to us is their normal
and they expect to still find wonder. Such a thing is not easy for everyone, though, so I can sort of understand why the presence of reliable at-will (combat) magic can tweak one's sensibilities.
Does spellcasting invoke awe or appear mysterious?
In general, no. Spellcasting is far too
mechanical in D&D. If you want spells (and spellcasters, for that matter) that actually feel mysterious, otherwordly, you need a game that processes spellcasting (and spellcasters) on a much, much more narrative level than any D&D edition ever has done. Dungeon World does a pretty good job of this, though I admit that my experience is colored by having awesome GMs who know how to turn a player-contributed world into something that feels like fables brought to life and myth being told at the table. It's not
impossible for D&D to do this, but you've got to make some very heavy modifications and alterations, to the point of nearly re-building the spells chapter, in addition to the way spells are acquired and employed--which, for 5e, is essentially rewriting half the PHB.
Do rituals invoke awe or appear mysterious?
If handled correctly, absolutely. It requires a deft hand and flexibility, holding the rules up as a thing to follow while also knowing when to allow shortcuts or substitutions. In other words, adjudicating rituals to maintain this perception of awe and mystery is very much like cooking. There are rules--but there also aren't rules. There are very good reasons why you use particular ingredients and cook them in a particular sequence, but there are also common substitutions, and being willing to experiment is necessary for getting the full enjoyment out of what you do. Following recipes, though, has to
fundamentally work, otherwise people will just give up before reaching the point where they're comfortable experimenting.
Do magic items invoke awe or appear mysterious?
As with rituals: if handled correctly, absolutely. Allowing items to just be dull stat-sticks with no other redeeming value or function is a waste of everyone's valuable time in my not-so-humble opinion. Whether you're in a game that expects, requires, or ignores the presence of stat-boosting items, the real meat of magic items is in the things that have great potential to influence the campaign but NOT along simple numerical lines. The reference to the ever-refilling chalice is a good example. My DW Paladin's sword, which has grown from merely a minor but important weapon of the Church of Bahamut into one of the most powerful artifacts in the known world, that has slain ancient vampires and raging dragons, that has been directly blessed by Bahamut himself, that has taken into it cores of power from
six of the great Towers of Magic (even though there have never been more than five in the known world at any one time)...THAT is an artifact that invokes awe and absolutely appears mysterious. The fact that, when "complete," it will become a Holy Avenger is almost a
footnote compared to its narrative significance; with the exception of our very first adventure, it has seen or participated in every major conflict our party has faced, from wrangling with bandits to
arm-wrestling the equivalent of wizardly demigods. And yet it is no less narratively important than the mundane, but preternaturally sharp axe the Fighter carries, the axe he stole from the (mind-controlled) Prince's armory a lifetime ago (though truthfully only a couple of years ago, in-universe), or the Cloak of Concealment that would make a normal person merely hard to see, but makes our Rogue
practically invisible unless someone specifically knows how to look for the right signs.
In summary: Much of "magic" can be
made to invoke awe or an appearance of mystery, but the two most active and dynamic parts of it (at-will magic and spells proper) actively oppose such characterization, because of how they have been constructed. A from-the-ground-up rebuild could make them...at least not
opposed to invoking awe and mystery, but achieving that is no mean feat especially for someone used to the system taking care of itself (as at-will magic and spells proper are expected to).
This brings up an interesting point though. While I agree that the blow back part is a major PITA, it does serve a purpose. Magic is
dangerous. You can be guaranteed that when the wizard player says, "I'm dropping a fireball", the entire table pays attention.
There was a chance, sometimes a pretty decent chance, of killing your buddy when dropping this spell.
Now, in application, yeah, it was far, far too fiddly. It's fine if every room is square or rectangular, but, as soon as you had an irregular shaped room, fergeddaboudit. My 15 year old brain wasn't going to calculate the volume of that cavern. Just wasn't going to happen.
But, there were a number of spells that worked on similar premises. Haste could kill you. Not likely, but, it could. It did age you a year. Teleport could kill you. Polymorph could kill you. Heck,
Raise Dead could kill you even deader.
This goes a long way towards my point of magic being sanitised. Magic in 5e rarely has negative side effects, and, because of the battle map (which 5e didn't start, true, but, does use), using magic is a lot more simply a math exercise. How many orcs can you fit into a circle with a radius of 20 feet?
And see, there's a really big reason for that. "The people have spoken," as it were. You call it "sanitization," but consider all the stuff spellcasting
used to have that it has slowly shed over time: hugely long memorization times (major inconvenience), risking serious harm by casting (random nastiness, and the "random" is equally important), having no control over what and sometimes
whether you learn new magic (major inconvenience), complex rules for how particular effects alter due to current situation (too fiddly/complicated/wastes time)...
Just about the only limitation on spellcasting, outside of daily resource expenditure, that
hasn't been removed or radically weakened is expensive material components (even 4e included them--for rituals). Concentration has technically stuck around in
some form, but I hazard to say that 5e Concentration rules are much more lenient than they were previously.
And none of that even takes into account the plain fact that the relative incompetence of frail cloth-wearing gits in (physical) combat has steadily decreased with time due to
other Quality-of-Life tweaks to the rules with time. They're still the bottom of the heap, most definitely, but it's been a long time since a Wizard could be killed by a housecat--and general public opinion seems to be that that was a
good move.
