It's quite a simple feat not to be ashamed at being white, but at the same time, not feel any sense of white pride. I for example, feel no shame whatsoever at being white, but neither do I feel any sense of pride about it.
Did you read the article cited in the OP? One of the main points there is that Whiteness is not defined by race, per se, but about about which groups have power, or are accepted into the social/economic/political power structures. I agree with this argument, which if true, implies that white pride is more or less equivalent to pride about membership within a privileged class, or a class that one believes ought to be privileged over other groups. Can you see why it might be a problem to go around publicly expressing pride at belonging to a privileged group? At the same time, I don't see any requirement that one feels ashamed for belonging to the privileged group. My membership in the group is purely the result of my birth; I've nothing to be ashamed about, although I can be and am ashamed at how many people within my privileged group behave.
Let me ask you this, if someone were to proclaim his/her pride in being white, what do you think the odds are that this person is a racist?
White pride, in practical terms, is a code word used by racists; it's not a concept used in the mainstream of 'white' society. Declaring white pride is a way of declaring your superiority over others. Can you now begin to see why some people might see this as problematic?
Let's try an analogy. Let's say that you were born into a 1 percenter family. Now, you need not be ashamed for being born into a very rich family, but still find it unsavory to go around publicly declaring your 1 percenter pride. Can you see how these are different things?