Given that you have already received some replies which I largely agree with I'm going to focus on some of the specific points of critique you raised.

The mystery and intrigue gets overshadowed by sexism,

I mean I can't really say that this isn't portrayed in the show, especially in the first few episodes, but I can't recall any instance of it being portrayed as a good thing. Quite the contrary actually.

jingoism,

The show is indeed rather militaristic, but given that the antagonists are a species of parasitic aliens with a god-complex

spoiler

(a plot point which gains more nuance in later seasons as well by the way)

I always saw this as a thinly veiled metaphor for armed resistance against the divine right of kings. So I wouldn't go quite so far as to call it jingoistic, although overtly militaristic is certainly a fair assessment.

characters that can be summed up on a postage stamp

As you already surmised this gets fleshed out a bit more later on, but stays more or less the same. Most characters, certainly the main cast in any case, stay rather archetypical with some character development happening though.

and plots and scenes that are contrived and clumsy.

This happens throughout the entire series unfortunately, but it varies a lot from episode to episode rather than from season to season. There are some rather interesting interpretations of common and uncommon sci-fi tropes throughout as well.

do they ever solve how the female token character is being treated?

Since you didn't exactly point out your problem with her portrayal I can only guess what you mean, but yes, I do think so. There are also other women joining the supporting cast (and even main cast in the last few seasons), leading to less frequent failure of the Bechdel test.

Does it ever stop feeling cheap and schlocky?

Not quite, but the first two seasons are certainly the worst in this regard, mixed in with most of the retconning happening to their content.

All that said, there is a reason the original show has 10 seasons at 22 episodes each, three movies, and four spin-offs, and if you can stomach early Star Trek TNG (or even TOS), you will probably enjoy at least the SG-1 series overall.

IMHO the first season is the weakest, second season is not great not terrible, 3-8 is the peak, 9, 10, and the two TV movies trail off a bit although still better than the first two seasons.

If you want to skip some seasons you should be aware that most of them have a "clip-show" episode towards the end that recaps the season and embeds them into the larger narrative happening in the background. I'd say the bad episodes are worth stomaching for the context though.

Atlantis spin-off is worth the watch if you liked SG-1 overall. Chronology is a bit weird though, SG-1 season 9 and 10 and Atlantis season 1 and 2 overlap.

Universe spin-off you can skip unless you got really invested.

spoiler

Universe ends without wrapping up its underlying narrative in any way since the show got canceled.

Haven't watched the animated spin-off, Origins was meh.

In conclusion, it's probably worth giving it a shot if you can manage to not take it too seriously.