Yeah, Trek ships have always moved at the "speed of plot." That said, I do feel like there are two major changes with Discovery versus earlier Treks which make it feel different.
One is that artistically speaking, they depict departure and arrival in a very different way. Instead of seeing a ship serenely glide into orbit, they jank out of warp instantly - often with a planet, asteroid, or another ship dangerously close to them. This is a relatively cheap way to add suspense - the Trek equivalent of a "jump scare" in some cases - but it's not necessarily more or less "realistic" than the classic beauty shots.
Indeed. TOS did have its moments of ships arriving all of a sudden, but those were done purely with dialogue, and expedited the plot by omitting dead time. The TOS movies went out of their way to show the serene gliding simply because they could. But TNG did not have that luxury, so it was back to the "Now we are here, orbiting X" dialogue, with even the "hero ship approaches red planet as viewed from astern" shot omitted from the stock.
DSC understandably does the best of both worlds. There's money and bits to burn for showing the arrival, but no time to show the gliding. So why not go for the scare factor as a bonus?
The other is I think it is the case that Discovery focuses a bit more on the destination and less on the journey. In Berman Trek there were plenty of episodes where the destination was more or less a framing device and some sort of negative space wedgie got in the way of the crew, causing the crisis of the week. Hell, I think in Enterprise's first season they were "heading to Risa" for around six weeks straight, but awful stuff kept happening. The ship here is more just a means to an end, but I think to a large extent the Spore Drive bakes that in. I mean, Picard had plenty of those slower-paced shipboard moments aboard La Sirena.
Destination mattering really is a Star Wars thing, an opportunity to show multiple locations within the single effective half-hour - another thing far beyond the budget of previous Trek TV shows. And yes, it's a natural result of the heroes having the spore drive. But there is a further level to travel in DSC, all-new yet not related to Stamets' invention in any way.
The thing DSC introduces is the ability of characters to appear and disappear within a single adventure, in what otherwise are bottle shows despite the multiple locations portrayed. Previously, we would get the mechanism by which a guest star joins the team, and then departs; say, there's a pickup at a starbase, or mention of a rendezvous usually complete with stock footage. In DSC, people come and go, not only being present when required by drama, but also suddenly explicitly being absent when either required to be absent, or even when not required to be present. That is, not just in the sense of "Sarek needs to be on Vulcan", but in the sense of "Sarek needs to be unavailable to talk with Burnham"!
This was not a luxury available to the writers of any previous Trek incarnation. But all of a sudden, characters can pop in and out, presumably by shuttlecraft, which for the very first time are suggested to have speed comparable to the big ships (in-universe first time; TAS has big shuttles do good speed a decade later). This is quite jarring when the role of Sarek is to be a mentor and/or foil to the main character, and her need for mentoring/foiling never goes away. But we get at least some consolation as regards Cornwell when in the S2 finale we're told she was a Section 31 bigwig after all, and quite motivated to move in the shadows.
S3 onwards, this activity is made even easier with the personal transporters...
Timo Saloniemi