It's of course not addressed in the games, but in some of the Halo books, it's stated that the Spartans have implants that suppress their sex drive.
It was a side effect of the augmentations, not a stated goal for them.
I can't comment on the Halo show in question, but John making out with anyone would be eyeraising at best, regardless of any circumstances. Leaving aside Cortana (which isn't really romantic - immensely close personal connection, yes, but not romantic, I'd argue), and the various female characters John's had comraderie with over the years (e.g. Kelly and Linda), the only time he's shown any potential romantic interest in anyone, ever, would be Melissa McKay. And if you ask "who?" then I don't blame you, since she only ever appeared in one novel, and hasn't even been mentioned since.
Is it impossible for a Spartan-II to be romantic? Not really. Maria started a family (though the fanbase tends to ignore her in my experience), and there's arguably a 'thing' between Fred and Vera, but if you're going to pair a Spartan-II with anyone, you really need to put in the legwork to do it.
(The Ones, Threes, and Fours are another matter of course.)
Like, I remember watching a Terminator tv-show and remember liking it, but a buddy of mine who was a die-hard fan of the franchise thought it was trash
I personally detest SCC as well. Which might be weird, since I genuinely enjoy every Terminator film bar T3 and BAT, but I've often wondered how much of SCC's fanbase were Terminator fans before the show aired.
Before anyone asks, no, I'm not trying to gatekeep, but SCC really doesn't jibe with Terminator for me. It's not the only piece of Terminator spinoff media that's had this problem, but it does share the same problem in a wider sense. The entire premise of time travel originally in Terminator was that it was a desparate, last-shot attempt by Skynet, and even T2 kind of breaks this premise arguably. But by SCC, we're in a full-blown time war between the Resistance and Skynet, with Terminators popping up everywhere, and where, in the midst of that, John Connor, future messiah of humanity...goes to school?
Also, it gets far too preachy IMO - technology is bad, scientists are bad, etc. Hackneyed as this is on its own, it doesn't even sync with the films that came before it. Yes, T2 did have things to say on scientific ethics, but it never advocated abandoning technology in of itself. Miles Dyson is conspicuously a good person, if naieve, not some amoral idiot.