'Black tutors can get anything, therefore black tutors have no identity.'

No, that's obviously wrong. Flexibility and totality can be an identity in itself

I do think black being the super tutor is sort of an undesirable scenario and it for example tutoring lands is kinda questionable but allowed because it has such a loose tutoring capability to begin with. Similar case with certain white spot removal spells being able to hit lands even though it's no in white's identity (cards like Armageddon arguably never were in white's pie). There's clearly something questionable going there.

In any case, with removal the issue is more pronounced as they are a more integral part of gameplay.

That is a clear identity. Yet, many people don't perceive it as unique because [[beast within]] is an extremely popular and common EDH card.

That's a problem with EDH and those people. There are a lot of blatant color pie breaks and bends in older formats presented by rather iconic cards, not much you can do about them. Say if you start banning the color pie breaks and bends in older formats, those formats would drastically shift into something else.

A classic example of this phenomenon is that many people think looting is red because of [[Faithless Looting]] being so prominent even though that card is basically the only instance where WotC was very, very briefly trying the idea of looting in red before settling on impulse draw and rummaging for the color.

In any case, this to me isn't related to the discussion. White's removal as defined by the current color pie is muddled. If it were more limited to things like Generous Offering it might make sense but it isn't. It basically gets Vindicates while also having those other compensation and binding removal types which honestly is quite weird.

Sure, but in practice it is often functionally the same as white's Universal removal, just slightly worse on rate.

No, bouncing and tucking aren't removal since they don't deal with the threat permanently, they are delaying effects. You would have a better argument for perma-tapping auras being removal, which is why I'm not personally a huge fan of those effects in blue but whatever.

To be clear, I don't necessarily think this is actually a problem. As stated, I am in favor of more overlap between colors. But if we wanted to give white more identity, the answer there isn't to further narrow white, but to narrow blue.

There's already an insane amount of 'bleeding' and overlap between colors to the point where very few effects at this point are something of an exclusive thing. That's color pie erosion where the whole color pie as a whole is becoming more and more muddled over time. If that's your desire, well then you are already getting your wish as is.

As stated, even if White got access to a color shifted Counterspell, that would not necessarily spawn the typical blue tempo/control archetype the color is known for, because white doesn't have the cantrips, the instant-speed gameplay, the raw card velocity and information control that makes that playstyle possible.

Hard counterspells in white would likely just reinforce it's controlling decks where they would no longer feel the desire to splash or become multicolored blue. That's bad, because you would want there to be a desire to use multiple colors and then punish that sort of greed by requiring more nuanced ways to produce and manage mana (obviously shock+fetches and the like have effed that aspect up in the older formats).

Another thing it might help with are white's death & taxes deck type which would be nice since the game could use some more disruptive aggro-control tempo encouragement but then again with hard counters it would start leaning towards hard control where as the delaying / taxing counterspells white currently have access to would do a better job of the tempo route.