Well, even though the first anime aired before the PSP game, the franchise was always planned as a media-mix project, with the anime and games being developed simultaneously, and the games are officially considered the "original work" and the standard around which the series is based. So if anything, it's the first two anime series that are taking liberties in terms of tone. It's not really comparable to things like Saki where the games are basically just spinoffs.
Why would the game be the "original work"? Is there proof of that somewhere that you have?
I mean, it started out with a radio drama, apparently, in '09, then the anime in '10, then the game at the end of '10. Why would the game be the "original work"? I don't know anything about the content in the radio drama, though, haven't seen anything about it.
It was meant as a mix-media project though, sure. On that note, another mix-media project (much bigger one) was .hack, and there too supposedly the game series was the first thing they worked on, but it's not that great, while the original anime (Sign) is fantastic, and turned out to be better than everything else in the entire franchise... so yeah, what really matters is what worked best, not whatever they were thinking of as the "original work". And especially in a case where the anime came first (as happened in both of these examples, I believe), whatever they were thinking during development, when it came time for release, the anime released first!
But they did such a good job with the madcap comedy tone (especially in the first season) that a lot of people, myself included, prefer the anime. I think there's room for both styles, and anything that gives us more of the four (not six, and DEFINITELY not two) adorable Milky Holmes girls is great, but if I had to pick one I'd take the anime every time.
I don't agree that both styles are needed; the Alternative version should not exist. I would also say that if new characters are actually done well, I wouldn't mind them. The problem is that these two new girls aren't done well, so far anyway. And they should be sub-characters, not main characters.