"Why can't we do X to solve the problem?"
"We can't do X because Y will happen."
You left out "... Y will happen
if the TARDIS stays in the pocket universe too long".
It's an important detail. You shouldn't leave it out.
You can't just turn around five minutes later, do X, and have the only justification be "Clara insulted the TARDIS"
Clara insulting the TARDIS had nothing to do with it. That's not offered as a justification for anything.
Meanwhile, throughout both these sequences, you have the psychic doing something. What, exactly? Nothing of relevance, apparently.
She opens a portal to the pocket universe. She does this 3 times during the episode (see below, I've labelled them P1, P2, and P3).
(Even if we ignore the fact that the TARDIS was there for longer than 10 seconds.)
Yes, we should ignore that. If your approach to film/television criticism involves the use of a stopwatch -- well, you should find another approach.
Okay, let's assume that Clara just didn't know that the psychic was going to pull it together so the second rescue effort was superfluous.
We don't need to assume that.
Because that's exactly what happens in the episode. The sequence of events was this:
Emma Grayling opens a portal (P1) to the pocket universe --> the Doctor goes through --> Doctor finds the Hila Tukurian and sends her back --> Grayling can't maintain the portal --> portal closes --> the Doctor is trapped.
Cut back to the mansion, where Palmer comforts Grayling, saying basically, "you've done enough, we rescued one" and Clara saying "we have to rescue the Doctor". At this point Clara runs off -- back to the TARDIS. Without explaining what she intends to do.
Next we see
two simultaneous rescue attempts -- each without the other knowing. Clara in the TARDIS (T1) and Grayling, after getting her... act... together, re-opening the portal (P2). Clara's attempt succeeds.
But since we already know that the psychic actually can pull people out of the pocket dimension, why put the TARDIS at risk a second time? It doesn't make any sense within the rules of the universe as they've established them in the episode.
We know Grayling can open a portal. Also that doing so is extremely stressful/difficult/she can't do it for long.
OK, so the coda/final scenes: the Doctor realizes there are two "monsters", one in the mansion and one trapped in pocket universe. He asks Grayling for a "favor", ie opening a portal (P3).
The Doctor goes through --> finds the other "monster" --> the TARDIS enters the pocket universe (T2) --> picks up the Doctor + monster.
The episode doesn't spell out why the Doctor asked Grayling to open the final portal (P3), but the answer should be obvious: to minimize the amount of time the TARDIS spends in the pocket universe, ie just to pick up, not drop off/pick up. It's not bad writing. It's the writer trusting the audience (or at least trusting the audience isn't out to get him). In fact, it's a very "gamer" solution. "Hey, can't we just have the psychic NPC open the portal? That'll save the TARDIS some exposure-to-the-pocket-universe time".
(whew... I feel like I just close captioned the episode for the interpretively-impaired).
I'm honestly amazed at the number of people trying to defend a blatant deus ex technobabble here.
The funny thing is there's no technobabble involved in the resolution. None. No adjusting the chroniton-phase modulation or even reversing the polarity of the neutron flow. They simply follow the "rules" established during the episode. Hell, there's barely a line of dialog during final scene, let alone any technobabble.
And because this post hasn't gone on long enough: any critique of the writing in Hide that doesn't favorably mention the real heart of the episode --ie, where it stops being a period ghost story and becomes a brief history of the Earth's whole history-- and the lovely, affecting lines given to Clara (and the Doctor's fumbling response) is lacking in the worthwhile criticism department!