I got my start with AD&D 2E, and all I can say is that from my experience, 5E doesn't really have "ubiquitous magic" on a scale substantially greater than 2E did. I will concede that at-will cantrips increase the presence of magic at low levels, but I don't think that is the case at mid to high levels. Cantrips don't feel especially magical to me, however, I don't find that they detract from the magical feel of other spells and magic. Even at low levels, 5E casters don't have an appreciably larger number of spell slots. The usage of spells with levels tended to be higher in my 2E games overall.
In 2E, the primary casters have roughly equal numbers of spell slots compared to low-levels in 5E (aside from the poor mage), but WAY more at high-levels.
Just look at these graphs. It's important to note that specialist wizards in 2E have equal or more spell slots than a 5E wizard at EVERY level except for 3rd level, at which they have one slot less. Arcane Recovery puts 5E Wizard over the 2E specialist until level 6, but it barely bridges the gap after level 6, and can't make up the difference past level 10. Arcane Recovery does give a special edge in that it allows slot recovery during the day. I also understand that preparing spells took alot longer in 2E, but in the games I played, there tended to be a good deal of downtime between adventures (which usually were two to three day ordeals at high levels), so it wasn't really a large constraint. Overall, I think as far as number of spells cast in a day, or on a round for round basis, 2E and 5E wizards are pretty similar.
A factor to consider when comparing Priest/Cleric slots between the editions is that while the 2E Priest certainly has fewer slots at low levels, they are able to narrow the gap via bonus spells due to a good Wisdom score. This is a relatively minor bonus, and likely only means an extra one or two 1st level spells each day, and maybe a 2nd level spell, but this is enough to bring the 2E Priest close to parity with the 5E Cleric. With a Wisdom score of 14, the 2e Priest has en equal or greater number of slots as compared to 5E casters at all levels except 3, where it is short one. Granted, the bonus slots are all 1st level spells, but again due to the auto-scaling nature of spells in 2E, that's not so bad.
Clerics have more options for dealing damage with magic in 5E than they did in 2E, but the ability to do damage via magic was certainly there in 2E. There are spells at every spell level in 2E for Priests that have obvious damage dealing capabilities, even aside from the Cause Wounds types, and they aren't bad. The following list isn't exhaustive omits a few spells that are direct damage types, and ignores the numerous summoning spells available to Priests that certainly can deal out round to round damage:
1st Level - Magical Stone
2nd Level - Heat Metal, Spiritual Hammer
3rd Level - Call Lightning
4th Level - Produce Fire
5th Level - Flame Strike
6th Level - Fire Seeds
7th Level - Fire Storm
In regards to Bards, I agree that they've generally become more magic oriented since 2E, but I find it to be an improvement. But just like with the primary casters, the major difference is at low levels when comparing the two editions. By high level, the 2E Bard has just as many spell slots as the 5E primary casters, though with lower level spells attained (maximum 6th level), but unlike 2E Paladins and Rangers, their casting level is not limited - it actually scales with their Bard level.
Paladins also have gotten more magical in regards to spells, but Paladins have always been very magical in general. In 2E, Paladins can detect evil, lay on hands, cure disease, and project a protective aura all beginning at 1st level (Turn Undead shows up at level 3). These abilities are mostly more limited than their current forms, but definitely make the Paladin feel magical without having spells. While the 5E Paladin gains spell casting seven levels earlier, the slots are fairly limited, and also used to fuel divine smite. My experience so far is that Paladins use far fewer slots for spells than they do smites. Overall, I find the 5E Paladin to feel marginally more magical than the 2E Paladin.
Rangers probably have increased in overall "magicality" the most. In 2E their spell casting was very minimal (maxed out at nine slots), and none of their other abilities seemed especially magical in nature. I find in 5E that though they have access to spells much earlier on, and significantly more slots, most of the spells are thematic, and don't feel especially supernatural. This certainly is subjective, so take it as you will.
All this being said, my overall feeling is that yes, 5E has a slightly greater magic presence at low levels, but by level 8-10, things start to even out. Go much above that, and I feel that 2E blows 5E out of the water with how ubiquitous magic is.