Solas himself asks the Inquisitor if the anchor changed her/his morals or spirit and an implication like that just doesn't sit well with me.
It sounds like that the Inquisitor would be under some kind of compulsion to do things he/she wouldnt normally do, or be able to accomplish the things he/she does.
This would really cheapen the character for me and especially the Solas romance, because it implies that Solas would have never looked at the little non-person twice without the guidance of a spirit through the anchor.

Yes, Solas asks Inquisitor whether Anchor changed something in them, but it was his last attempt to rationalize his stance that hardly anyone could have a deep connection to the Fade/spiritual side or that it would be so strong.
He drops the idea in the very same conversation though - he's still baffled by Quizzy, but you can hear (in case of Lavellan) him asking "if the Dalish managed to raise a person with a spirit like yours, have I misjudged them", meaning that he accepts that Inquisitors were either born "speshul" (or they managed to retain or reach their "speshunless") despite overwhelming odds.
I don't think that if he thought that it was all because of the Anchor, he'd have accepted anyone as a person and said at the end of Trespasser that Inquisitor has shown him that he was wrong.
...And honestly, I'm still confused why people think having a strong spirit (especially considering that likely everyone has one/is one), even one with specific quality, in any way "cheapens" the personal accomplishment and growth of good Quizzies - having one, just like having an Anchor, doesn't mean that they're "speshul" by default, in fact they're likely more prone to corruption than your average Joe.
Most people with stronger connection to Fade are, after all. Mages, Dreamers, etc - all of them have a heightened risk of being either broken or possessed, either because of demons trying to get into their head or the world set against them.
Add to that an allure of power and prestige bestowed by the Anchor and title of the Herald, and it becomes clear how easy it is for Quizzies to falter. And we've seen that this can happen - we don't have an option to have downright "evil", but Inquisitors can be nasty, arrogant, immature, short-sighted, bellicose, gullible or just plain unimpressive.
Therefore IF Quizzies indeed have a particularly strong spirit or naturally stronger connection to the Fade and it leaves them more vulnerable to fall, wouldn't their personal growth be even more impressive?
Personally I think that it would.
They might have had almost the whole world against them, they might have felt isolated, threatened, misunderstood or singled-out, yet they managed to persevere. The world didn't break them, even if it should (and Solas definitely thought that it would), in fact it perhaps managed to 'temper' their spirit and character so long as they didn't give up on it and its people.
Therefore they're 'speshul' not because of the Anchor, not because of the possible 'special mission from fate' and not because they might have been born "speshul" (not just in terms of spirit, but perhaps as a mage or member of nobility), but because they wanted to. Really, I'm not sure how that in any way this could "cheapen" their character.