Even antiquated, marriage is available.
Naah, marriage is just one of many modern and equally viable options in TNGville. TNG, clunkily or otherwise, showed single and two-parent families, albeit often nothing more than as plot devices but that's a different story (or numerous ranging from mostly seasons 1 and 5)... and/or cheap lame soap opera given how Bev and Wes acted to each other. Just nothing more extensive than those, like a family of 5 similarly-aged adults working together and raising children for whatever reasons... if this is the sort of plot fodder people want, writers can have a field day with it and go the extra 5.88 trillion miles.
Especially when TOS talked about crewmembers marrying a few times, wouldn't it be interesting if sci-fi penciled in and postulated how within 78 years such a "modern" paradigm could be told effectively, whatever "modern" is as TNG had far less of a clue than any of us had in terms of what's optimal or ideal or even acceptable or what not?
I didn't touch on DS9 and the Ben/Jake situation. By then they had some inkling of a single parent household that would function and did it and often without the often laughably bad melodrama that Wes/Bev got into. Or VOY but if it's a small ship and they start breeding like Tribbles... would a Tribble merrily coo when placed next to a Borg or would it squeal like it does when transported onto the lap of a Klingon? Like the number of licks it takes to get to the center of a Tootsie Pop, we may never know...
If Picard was that concerned about having children around, possibly he should have married the woman instead of just doing a one night stand.
If that episode is to be believed, but there's a more fun tale to be told:
Picard often said children made him uncomfortable and awkward. He would not want kids. Given then umber of times he hyper-strategized in advance, especially in season 3 on so many other issues affecting far more people like when dealing with the Sheliak or Romulans, it is highly unlikely he's going to go Kirking around the galaxy so haphazardly the way Riker was - and yet TNG didn't explore all of Riker's kids knocking on his door, but I digress. Picard would more likely go to Doctor Crusher and ask for a duffel bag full of contraceptives and prophylactics. And to make balloon animals with since he not only had a triple fisherman's knot vasectomy while in Starfleet Academy, he was also rather annoyed whenever women approached him because of his big Horga'hn sitting there while he was reading his dirty book on Risa.

"The Naked Now" may have hinted at some restrained emotions and wanting to do the jiggy deep down (pun not intended), but "Captain's Holiday" explaining a lot more and a lot more quickly and with a lot more panache. Then came "Q'Pid", which wrecks everything by having Q goad Picard over his now having big feelings for her, apparently. And as "Q-Less" proved, she had cooties Q de-cured her of and it's for the best Picard kept his swimtrunks zipped at the time...
In other words, that season 7 escapade redoing to good old Ferengi revenge trick to reveal an alleged kid... I found it hard to believe at the time.