What I meant is that power source interacts only very weakly with the actual game mechanics. For example, there's no Antimagic Field, Magic Resistance or Counterspell to worry about in 4e. Healing is healing, it doesn't matter if it came from a martial, divine or arcane source. It's almost always just fluff.
By the way, the application of the Arcana skill you just quoted specifies that the magic effect must not be generated by magic item, power or ritual. Why do you think they added that bullet point?
No, I only copied the section which was relevant to a power effect. But if you want the full description including other effects, like rituals, zone or actually just sensing the presence of magic, I've copied it below.
Note in particular about sensing magic that you sense the power source, if any. So no, the power source is not just fluff, but clearly, although the information is not repeated in the rules (4e being efficient in that respect as with many others), the detection and identification of magic, including the power source (which is critical, because some power sources are clearly identified as magical and others as not magical) is part of the core rules. So I would not qualify this as "weak interaction", it's pretty well codified straight in the PH and in two sections referring to each other.
As for antimagic, although I agree that it does not exist in the core rules (although the support is obviously there, see above, it's easy to make some power sources impossible to use in some areas), I think this is a failure of 4e to lack support for a feature which is so much a part of the genre (Wheel of Time makes a lot of use of it, Sanderson in general, I'm sure I can find other examples if pushed). It is actually well part of a lot of power curves in books in particular, because once the characters become proficient in their magic art, the antimagic is one way to bring the characters back to basic. It might even be a slightly overused trope, but D&D is a lot about this anyway.
It's less true about Magic Resistance but honestly, I again think it's a bit sad. Not only is it a long-time trope of fantasy and D&D, but it also makes for interesting play. I know that 4e does not need rebalancing, but it's also one of the game changers at high level in every other edition of D&D, rebalancing things in favour of non-spellcasters, and making the game more varied.
Counterspell is even less important, but it's widely used in fiction as well, and since we are speaking about this in this thread, I also prefer a system where I can model fantasy tropes easily.
Identify Conjuration or Zone: Minor action.
✦ DC: DC 15 + one-half the power’s level. You must
be able to see the effect of the conjuration or zone.
✦ Success: You identify the power used to create the
effect and its power source and keywords.
✦ Failure: You can’t try to identify the effect again
during this encounter.
Identify Ritual: Standard action.
✦ DC: DC 20 + one-half the ritual’s level. You must be
able to see or otherwise detect the ritual’s effects.
✦ Success: You identify the ritual and its category.
✦ Failure: You can’t try to identify the ritual again until
after an extended rest.
Identify Magical Effect: Standard action.
✦ DC: DC 20 + one-half the effect’s level, if any. You
must be able to see or otherwise detect the effect.
✦ Not a Power or a Ritual: The magical effect must
be neither from a magic item nor the product of a
power or a ritual.
✦ Success: You learn the effect’s name, power source,
and keywords, if any of those apply.
✦ Failure: You can’t try to identify the effect again
until after an extended rest.
Sense the Presence of Magic: 1 minute.
✦ DC: DC 20 + one-half the level of a magic item,
power (conjuration or zone), ritual, or magical phenomenon within range.
✦ Area of Detection: You can detect magic within
a number of squares equal to 5 + your level in
every direction, and you can ignore any sources of
magical energy you’re already aware of. Ignore all
barriers; you can detect magic through walls, doors,
and such
Success: You detect each source of magical energy
whose DC you meet. You learn the magic’s power
source, if any. If the source of magical energy is within
line of sight, you pinpoint its location. If it’s not within
line of sight, you know the direction from which the
magical energy emanates, but you don’t know the
distance to it.
✦ Failure: Either you detected nothing or there was
nothing in range to detect. You can’t try again in this
area until after an extended rest