Why does the Supergirl TV series go out of its way to not say "Superman"?

Because they want the show to just be about Kara:

“Our prototype is the way the president is seen on Veep,” executive producer Ali Adler tells EW, referring to the HBO drama’s clever way of mentioning, but never showing POTUS. “It’s certainly [inspired by] so much of what Julia-Louis Dreyfus’ character goes through. Ultimately, this is a show about Supergirl and we really want to see it through her lens.”

Because Clark Kent is a Clark Bore:

ADLER: I think there’s going to be our learning curve and then a teaching curve. We can’t just paint with two colors. I think Kara is very dimensional, which is exciting for us. Superman — at least in the comic books — is, to me, less interesting, because his range of emotion can only be... he’s an autonomous guy. So because Kara has friends, and this wider range of emotion, she has a different origin story, and grew up on Krypton, she knows two worlds, she’s just got a wider range of emotions we get to use.

But even at the beginning, while saying they want to focus on Kara, they always wanted to have Clark on a larger scale:

Producers have previously stressed this is Supergirl’s story, but they left open the possibility of seeing the Man of Steel on the series. “It will always be a dream of ours to be able to have Clark or Superman on the show,” Kreisberg previously told EW ahead of the season 2 renewal. “Hopefully with continued success, that opportunity will one day present itself.”

And they succeeded:

“Greg [Berlanti], Ali [Adler] and I are beyond thrilled to welcome Clark Kent and his slightly-more-famous alter ego to the world of Supergirl,” says executive producer Andrew Kreisberg. “Superman will be appearing in the first two episodes of the new season we cannot wait to see who next dons the red cape!”

The issue, along with budget and theme conflicts, was likely CBS, the network the show was previously on. Logically speaking, CBS is also the likely culprit for why the Flash was not allowed to talk about the Supergirl/Flash crossover. The CW is much looser with the reins, they talk bigger chances and allow the shows to flourish, while CBS is a legacy network set in its ways. Without the CW taking over for Supergirl, CBS was ready to cancel the show via non-renewal.

The very same time that Supergirl was confirmed as moving to CW for renewal, we get confirmation of a multiverse crossover between the 4 shows. A few weeks later, we get confirmation of Superman. It's no coincidence.

So the answer is really a simple "Executive Meddling".

They are clearly allowed to mention him, since they do so at least once in the pilot, and several more times during the course of the series. They also make sure to work in every one of his usual comic book nicknames (which people in-universe use to refer to Kal-El pretty frequently).

As far as I know, the writers have never specifically explained why they went to such trouble to avoid the "S" word. However, they've been pretty open about the fact that they intend to keep Superman out of the show as much as possible:

And I think the reason why Superman’s not in the show is because it’s not about him. And it’s her show and it’s about her and he is just sort of a shadow in the back of her psyche, and I think that’s the right way to portray him because that’s how she experiences it. ^src

It makes sense, then, that they don't want to keep bringing up Superman by name, because that just reminds the audience that there's another, "bigger" superhero running around this world. Obviously, Kal-El is going to be a big part of Kara's life, but in this show, he's just a background character. Most of the time they're going to refer to him as "Kara's cousin" to reinforce for the viewers that the show is about Kara, and we're seeing everyone else from her perspective.

After the pilot it gets a lot less clunky. In fact, in the changed intro, he's mentioned by name in every episode: something like "you may know his story" was changed to "..and become Superman". And he's usually referred to as either Clark or Superman throughout the rest of the season, depending on who's mentioning him.

I think in the pilot they really wanted to just obliquely refer to him. And then there was a lot of feedback saying "fine, but he's got a name". And as the show grew stronger in itself, it was less of a threat to just mention his name.