You are a little off here. When the alerts have a high repeating rate, ONLY THEN is the user presented with that option. This is to prevent infinite alert loops which would block the browser.
Yes, you will control the appearance. But replicating the scripts execution stopping the native alert features, that's a thing you yourself have not solved in your replacement you linked to, a few posts up.
No one uses alert to debug these days, or even reasonable distant times, it's a toy tool, it always has been.
No, that's precisely the reason for not hijacking the alert. As any native can be easily hijacked, there's no guarantee that your replacement will take precedence and will be seen at all by your visitors as you intended - another reason not to hijack them.
Hijacking natives also spells out questionable intents from the web site.
And that overriding would make it for a half ass job which would cost any real developer its job. Do it properly, learn to do it properly, or step aside. That's my opinion.
You seem to be going around in circles with this. I'm going to stop here.