I think I'll answer this in two separate answers, so as not to conflate in-game issues with real-world issues.

To address the issue in the game world,. it's not quite as easy as issuing a statement of support. After all, It's already known that the church supports Konsi, a lot of people won't see the statement, and nothing they could say would come close to measuring up to open recognition of her deeds by the Open Lord...

The issue isn't really about Konsi's character at all, the issue is that there's an opportunity for opponents of the church to run a smear campaign, and there's no shortage of people who might want to do that.

Agents of Beshaba will not care if the majority of people believe them, and won't be trying to convince everyone. They just want to publicly allege corruption, throw in some doubt and confusion. They want to give ammunition to those who want to be convinced that the church is bad to back up those views. Give them ammunition to reinforce their existing beliefs.

There are plenty of people who dislike how much power the Tower of Luck has, especially over the gambling industry in Waterdeep. Some people don't want to pay fees or employ members of the tower to oversee games, it's almost like guild dues. Some people want to run crooked games, but find it difficult to find marks when all the reputable games are sponsored by the church. Some people believe that it's not the place of the church to be doing any of this, the size and power of the Tower of Luck isn't at all consistent with how the faith is normally represented - there are theologians who argue that Waterdhavian Tymoran faith is a heretical sect.

Some of the nobles who lost in the tournament might have a bit of a bruised ego from "losing to a goblin" - if they can cause a controversy about the match, claim there was a fix they can convince people around them, and themselves that they didn't "really" lose.

And... of course, those people who just absolutely hate, hate, hate goblins (of which there are many) will be inclined to believe the worst, no matter what anyone says or does.

If such attacks were made, would it seriously damage the church? Probably not - but it'll cost them a few people here and there. Worse, it could form the foundation of a sustained assault on the position and power they hold in Waterdeep, and it's the job of the church's leaders to protect that.

I imagine, if anything does happen, there will be more specific details in future comics...