Star Trek: Discovery (2017)

In A Matter of Time, discussion about what the 22nd century was like and Riker said that warp coils hadn't been invented yet. In the TNG episode Rasmussen, "I suppose the warp coil. Before there was warp drive, humans were confined to a single sector of the galaxy." - which indicates that warp coils are necessary for warp drive.

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Yeah, there's been a lot of discussion over the years in even nerdier circles dissecting that one line, clause by clause. The general gist goes:

• Various non-Human species had FTL propulsion well before us, so we regard Riker's comment as being from a Human-centric perspective, oddly chauvinistic compared to the rest of his portrayal across the series.

• While it's true that before warp drive, Humans were confined to a single sector of the galaxy, everything else points to high-sublight drives in the last decade of the 20th century and the development of warp-1-capable ships by mid-to-late-21st century, so that sentence is regarded as expanding indirectly on the prior sentence, as...

• ...Much, much discussion, which I'll condense here, to the effect of there was reactionless drive before warp drive, as the Botany Bay model had no exhaust openings on the back faces of any part of its structure; Cochrane likely didn't have antimatter, dilithium crystals, or exotic metals like the "verterium cortenide" the TNG Technical Manual references; and that it was liklier that Cochrane extrapolated from existing technologies rather than conjuring his FTL drive from utterly whole-cloth. It was charted out that if it was a straight logarithmic progression from warp 1 in 2063 to warp 8 in 2245, that'd be about one integral warp factor per 23 years, with an error margin for setbacks and breakthroughs increasing or decreasing those intervals. But by that, they'd be hitting warp 3 in the early 22nd century. If fission gave way to fusion gave way to antimatter, if refinements and tweaks revealed the energy-consumption thresholds between warp factors... We figured it's the most probable course that the Phoenix had some sort of pulse-fusion reactor that could push an early version of his engine just past the warp threshold, and that as they studied and developed things over the next few decades, they were able to turn it from an incremental advance over the previous high-c-fractional drives into an actual warp drive, as we tend to think of it.

So we generally parse Riker's comment as indicating a view of history that holds that Cochrane invented the technology in the 2060s, but that it didn't become a viable quantum leap over what had existed before until the turn of the following century. Further things like the Phoenix being called a "warp test ship", Cochrane being described as "the [Human] discoverer of the space warp", and the Bonaventure being described as "the first [Human] ship equipped with 'warp drive'" we interpret as implying a gap between the technological breakthrough and it being refined into a viable, reliable drive system. Roughly equivalent to our often-problematic development of the initial V2 rocket technology into something that could take us to the moon without blowing up on launch.

You mentioned TOS' Where No Man Has Gone Before which strongly implies that the Valiant is a pre-warp ship,,, hence this:
Spock: Decoding memory banks. I'll try to interpolate. The Valiant had encountered a magnetic space storm and was being swept in this direction.
Kirk: The old impulse engines weren't strong enough.

The implication is there. Is it concrete proof? No. But, the intent seems pretty solid.

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Mm. I'll throw these quotes from downthread in, as they're germane:

The Mad Professor said:

1) The shuttle. Dear GOD that thing was beautiful. Disregarding the warp capability (something that shuttles on the Enterprise did not have), the design was nice.

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robn1 said:

They had warp nacelles.

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And the Romulan ship in "Balance of Terror" was described as being powered by "simple impulse", and the TOS shuttle blueprints show the nacelles as being "impulse pods" or something similar. Over the years, the attitude in the Treknical community has shifted from "impulse = sublight" to "impulse = one type of power generation, of lesser capacity than antimatter". That Romulan ship likely did not take decades to traverse the distance from its base to the Federation's Neutral Zone outposts, nor did it take years to get between outposts. The TOS shuttle was shown to be crossing interstellar distances more than once. We've come to figure "impulse" is a form of pulse-fusion-powered continuum-distortion drive (hence the "spacetime driver coils" in the Enterprise-D's impulse drive) that lower the ships' inertial mass and create distortion "ripples" that the ship rides. The exhaust ports are pointed aft to vent waste products along the ships' usual direction of flight, rather than being reaction-drive propulsion like a rocket.

Impulse or similar is fine for sublight propulsion, and probably enough to push a ship into the low warp factors (maybe as much as warp 2 or 3). That would let a stealthy Romulan ship or a Starfleet shuttle cross nearby interstellar distances in reasonable times without needing something as powerful and volatile as antimatter to do it. And on antimatter-fuelled ships, impulse drives can add a power boost to the FTL drive (in TNG's "Conspiracy", an impatient Riker orders an acceleration to warp 6 and Geordi responds with "Aye, sir. Full impulse." The already-Engineering-inclined LaForge used the ship's impulse drive as an acceleration assist, rather than putting the strain on the antimatter reactor -- a bit like newer hybrid performance cars that, in one operating mode, use the electric motors to assist the gasoline engine on acceleration and to smooth gear changes).

JD said:

]In TOS Metamorphosis, Cochrane is a human from Alpha Centauri - later he's from Montana and had moved to Alpha Centauri. The original intention was that Cochrane was the originator of warp drive - not Earth warp drive, but for the galaxy - which makes Star Trek and the Enterprise's mission all that more sensible (at least to some).

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In TOS it was implied that Vulcans had warp drive before Humans did, by at least a couple hundred years. This, pieced together across several episodes' dropped references. There were other things, such as Spock being the first Vulcan in Starfleet, then, later, the distinctly-not-Vulcan-named Intrepid being crewed entirely by 400 Vulcans. It was a Starfleet ship, though, as it was on Commodore Stone's status board at Starbase 11. The overall pattern is that Starfleet has been traditionally a Human-only or Human-majority organization that is only contemporaneously becoming more diverse at the time of TOS. This seems to be more participatory on the parts of the non-Humans than exclusionary on the part of the Humans. So I and others take a lot of the dropped data in TOS as being from a Human point of view. Including, as I mentioned above, Cochrane's status as discoverer of the space warp.

Also, because we were sending out sublight sleeper ships roughly 70 years before Cochrane broke the light barrier, I'm fine ignoring the retconned backstory Enterprise gives him and am fine having him be born on an Alpha Centuari colony, make a first near-warp test flight to Earth, and get stuck there due to WWIII. There's about a twenty-year error margin in the character's age, between all factors, but even at his oldest, he'd've been born after the turn of the century. Reference my earlier gripes about the showrunners not knowing/caring about their own lore. A good writer can (and often enjoys) work within the confines of established data. A sloppy one ignores it and just tells the story they want to tell, regardless of how much it might contradict in the process. More and more failure-to-research has crept in over the years, but it's still possible to compensate for most of it.