No, it's in multiple places that aren't all "from the Bard's point of view." The "Bards say..." part is about the exact nature of the power they're tapping into, not about the idea that they're tapping into
something.
What's the difference between "drawing on the power" and "magic"? I ask because the idea that a warlord's words draw on some power outside of the warlord's being and personality has not met with approval when it was proposed before (see
this thread; note that "inspirational healing must be explicitly non-magical" is the second most popular choice by a narrow margin.)
Magical support of bard's and clerics could well be reference to their actual spells. Come on, now.
Bards are magicians. True. They cast spells.
Bards are masters of words, songs, and the magic they contain. Also true, I guess. Though that's now speaking about the setting, and that's going to vary from campaign to campaign. The PH doesn't tell me how the world was created.
The vignette is likewise in-universe. Which may be cool for most universes, but can't be taken as law across every one.
But you notice there's nothing explicitly magical about Bardic Inspiration in its own entry.
So what's the difference between drawing on the power and magic? When we talk about wanting a non-magical support class, we want to be able to produce effects without reliance on spell slots, without those effects being dispelled by dispel magic, or negated by anti-magic zones, or coming with any religious or arcane baggage like demanding deities, pact-patrons watching over your shoulder, bloodlines, colleges, or cabals of wizards.
If you need "because magic" to justify stuff, cool. Have at. I don't. I'm not hostile to it in the same way I'm not hostile to the idea that giants exist in D&D (even though realistically they'd collapse under their own weight). But giants don't collapse under their own weight. Dragons fly - when they shouldn't be able to - and you can either shrug and go "cool, fantasy" or you need "because magic."
So alright, it's all magic. Everything. But bardic inspiration doesn't work like spells. And what a warlord doesn't want to be is a spellcaster. Even a barbarian who takes the totem path and gets one or two spells isn't a spell caster. What he does may be magical, fantastic-al, but it isn't magic (a la the casters).