I agree with this excellent suggestion, for a couple of reasons.
I do enjoy stories which move other regular character's stories forward, such as Nia's story this year, and Brainy's growth as a character too, or the ups and downs of James' and Lena's off-and-on romance, and Alex's side-tracked and then back-on-track story about her desire to adopt. I don't find these distracting at all, and the personal aspects of them adds depth and humanity to their characters too.
As for Dreamer; I think that it would be great for Kara to loan Nia the Legion ring that Mon-El gave her as a keepsake before he left to return to the 31st Century. If Brainy uses his Legion ring to fly (among other things), they it would add a great new dimension to Dreamer's abilities in her super-hero role, and be a nice friendship/bonding gesture on Kara's part too.
However, I do agree that too much time is being wasted by unrewarding side-stories which don't really go anywhere; such as Manchester Black and his 'Elite', which eventually fizzled out, or J'onn's obsession with Manchester, and how it briefly led him back to Mars. I would much rather have enjoyed seeing J'onn evolve more in his away-from-the-DEO role.
And although I enjoy a sense of reality to what is obviously a fantasy show, I find that this show seems to have become much too politically-oriented, and find myself wondering where they are hiding the podium for the next preacher to make their social-commentary speech from. Some reality is good, but isn't a TV show supposed to be primarily about entertainment more than politics? At times, it certainly seems as if the writers are writing for their own pleasure, rather than for the pleasure of their viewing audience.
As for the suggestions by earlier respondents as to how to improve the human side of 'the gang' in Season Five, I like most of what has been put forward so far by others. For example, having Supergirl making public appearances around National City would certainly give her more of a personal presence for the people in the city.
And seeing as how Kara Danvers saved Supergirl's bacon recently, it would be nice for Supergirl to acknowledge that publicly. Perhaps she could be holding an impromptu open-air press conference sometime after doing something 'super', and Kara could be one of the reporters asking her questions afterwards. It would be a great way for Supergirl to thank Kara for not giving in to the lies that were being spread about her, and for believing in her and writing the story which set the facts straight, exposed Lex, and got the puppet-President Baker removed from office. And having countless other people seeing Kara and Supergirl in the same place at the same time certainly couldn't hurt either.
As for the 'team' going out together; I totally agree, but as their real selves, not as their super-personas. I really enjoyed all the times that they hung out together in the alien bar in seasons two & three, and missed that in this past season (although it was good to see the bar's owner, Al, show up in a few episodes this year). Going to a hockey game together, or a football game, or baseball, or basketball, would be great to see, assuming that National City has teams in the NHL, NFL, MLB, NBA, or MLS.
These kinds of group activities wouldn't have to take much time away from whatever the story is for that week, any more than their bar-nights did in previous seasons, and I agree with the others in this thread that it would contribute greatly to the friendship and bonding that this group of friends and clandestine super-heroes shares with each other (assuming that Lena and Kara will patch things up over her being secretly Supergirl, and that Kelly will eventually become an insider to everyone's secret too).
And it would be really great if Alura could visit Earth from time to time, not during emergencies but just to visit with, and get to know, Kara's family and friends. It would also be nice to see her and Eliza hitting it off together, especially if Alura could attend the Danvers' family Thanksgiving and Christmas dinners.
So thank you, Stewart Tick, for a great idea, and I hope that others will contribute their suggestions towards this topic.