To be fair to the later series, I think the *ratio* of time travel episodes within the series remained didn't increase - it may have even gone down. There were a lot more episodes later that involved time travel, but there were a lot more episodes total. Here's one person's summary:
https://www.higgypop.com/news/star-trek-time-travel/
That's 5 out of 79 in the original series, 10 out of 178 in The Next Generation, 10 out of 176 in Deep Space Nine, 11 out of 172 in Voyager, and 12 out of 98 in Enterprise.
That's a large number of episodes, but the vast majority of episodes have no time travel - so I agree that retconning them out isn't that difficult. Taking out transporters would be a lot more difficult, as they are used in nearly every episode.
Problem is that Enterprise skews this in a roundabout way as theres that whole "Temporal Cold War" arc. And Discovery has been doling out the time travel elements left and right. Picard too. Prior to that I'd say yes. It was not as prevalent or in some cases was time travel only in the most roundabout or superficial manner.
It even pops up as a story arc in the Star Trek early NES/SNES console games. And its all over the ST MMO.
As for Transporters. Those could actually also be easily removed and replaced with shuttles. A few episodes would need some sort of fast emergency pod like a suped up workbee for the rare few really close calls. Or other tech to extricate characters from whatever problem, or to get them into whatever problem.
Just a matter of how many hoops you want to jump through to get what you want. Much akin to say removing magic from Shadowrun. In some ways it impacts the setting not one bit as it's always treated the magic as an afterthought even when its supposedly prominent to a story.
I don't think removing transporters would change much at all to the overall story - except we'd lose Mirror, Mirror (which by itself was an interesting episode like Remember Me, but got out of hand with the mirror universe episodes in DS9) and Second Chances - since a lot of the time the writers had to find some reason that transporters didn't work to maintain tension.
That last part is why the Modiphius version is a heavily narrative game. In traditional games, people expect their gear (magic, tech, or magical tech) to have predictable results and not to go on the fritz for story reasons. In the narrative games, you always expect the gear to relatively unimportant unless you metacurrency it into being the most important thing.
I think that by plagiarizing getting your inspiration from sources other than ST you might solve the transporter problem without getting rid of it altogether.
In Ringworld humanity has transporters, they are closed cabins and have a distance/speed limitation.
I don't remember the exact number but you were limited to certain ammount of kilometers, you needed several jumps between different cabins to manage to go around the globe (earth).
The speed limitation meant that both cabins had to be at a constant distance from each other and travelling at the same speed in the same direction.
Later the Pierson (titerotes in spanish sorry) are shown to have an open system which only requires a transmiter pad + receiver pad and has greater reach and tolerance for speed differentials.
ST transporters already have a distance limit, but it's roughly 40-50k kilometers for Federation (and most other Alpha and Beta powers). OTOH, the Dominion has transporters that can beam people from nearby star systems...but they rarely make full use of it.
Yeah, much less for ringworld humanity ones, plus the need of TWO cabins and the speed limitations.
This means you could still have one in a shuttle but the party needs to get to it and then beam themselves (one at the time) to the mothership assuming it's in a stationary orbit around the planet.
Instead of ST matter replicators Ringworld food kitchens. The latter need to be re stocked with organic matter and can't replicate a wellington steak, they just produce nutritious bars.
In the mother ship you still get more complex kitchens that do replicate more complex foods, but still need the organic matter source. It's a little less unobtanium or handwavium or whatever.
But fuck time travel, that shit only gives me head ache.