I really don't think Trek can have both a TV series and movies on simultaneously anymore without it eventually being a repeat of what happened last time--it gradually loses its novelty, mainstream audiences get their fill of it and tune out, and people start talking about "franchise fatigue."
Why does it have to be the same? I'm envisioning something very different in content. The similarity would be that Starfleet is always present, and the production design elements would also be similar. You'd see the same visual cues in uniforms and sets, not so glitzy on a TV budget but still respectable.
For the movies: Kirk & crew have a wild, colorful, violent, zap-pow adventure for two hours every three years.
For streaming: a different set of characters, Starfleet and otherwise, in a more plot and character focused context, with more talking and less action than the movies and a plotline that unspools over time in a serialized fashion like cable shows.
Maybe they're colonists on some alien world with some kind of unknown danger that only emerges over time. Maybe it's a starbase that follows the complex workings of Fed diplomacy and peacekeeping. There are lots of premises that can be developed for this format.
The wide-open possibilities of streaming mean that several Star Trek series could be developed, aimed at different audiences. A Starfleet Academy series for the CW type audience. A cartoon aimed mainly at kids. Eventually even more esoteric topics, such as a Klingon-focused show might be viable once the franchise is built up.
Think LOTR vs Game of Thrones. Same basic genre, but different approaches. Both can be successful without stepping on the other's toes.
But "Event Television" can still bring together a family to view together: Down Under, "Big Brother" came back after a "franchise fatigue" hiatus - not exactly "quality" family viewing, but it rated very well.
Whatever the approach is, it's gotta work in America first and that won't work in America. The time for that is past. The family views TV on different devices at different times. The parents are watching on their iPads while commuting and the kids are watching on a laptop in their bedrooms while pretending to do their homework. Sports will lure dad into the living room to watch on a big-screen TV.
Also, Big Brother is a reality show and that's very different from a niche genre like sci fi. There are a few genres left that are still mass market - reality, cop shows, sports, some sitcoms. Those might still attract family viewing, but sci fi is at the other end of the spectrum of mass-to-niche tastes.
All of these shows have been very popular with mixed-age households - and these are attractive because a cross-section of advertising markets are available, and related "TV tie-in" products sell well because one family members buys it for a different family member.
You're assuming Star Trek will be ad supported rather than subscription based, which is definitely not my assumption. I think subscription is the future for a niche genre like Star Trek.
Also, advertisers like to be able to target their audience in as focused a way as possible. If they can serve ads to mom while she's commuting and dad watching sports on the big screen and junior watching ads on his laptop, they love that. They know exactly who they are reaching and pay only for the eyeballs they want.
Now that the internet is letting them target like never before, and only pay for results (pay per click), they are going to be increasingly critical of unfocused TV advertising where they don't get direct feedback about who precisely is watching and there's no way for anyone to click on a TV ad and prove that it was effective.