Consider if this idea did sell as a series. Imagine a 13 episode season wherein each season as one year of the 5-year mission and the thirteen episodes are high points of that year. That's equivalent to a major event happening about every four weeks on average. This alone gives things a bit more sense of realism rather than accepting something exciting happening about every two weeks. Now you can only hope and work hard to make your show last at least five years.
Another thought could be to plan your 5-year mission to play out over three seasons, which makes things seem even more realistic and allows even more time for these pesky mundane routines between high points to deal with not-so-interesting operations. Anyway, assuming you get renewed to can plot another 5-year mission to run over the next two or three seasons.
Now, let's assume we've managed to keep things going for 5-6 years. Consider if we tweak the show particularly character wise. You promote and/or reassign characters and bring in a new crew for a third 5-year mission. Enter Picard and company, also slightly reinterpreted as was done with the TOS characters. You're able to segue seamlessly into your "next generation" without losing the general feel of your show, and you're saving money by bringing in a new cast rather than having to pay more for the original. You give the show a creative boost as well as establish new and different character dynamics. You're also saving money because you already have standing sets and such which you might only tweak a bit. Another consideration could be to update the writing staff to help evolve the show.
Having Picard commanding this Enterprise after Kirk could be more like Picard commanding the Stargazer rather than the E-D because there would be no families aboard. Another option would be for Uhura (as First Officer) promoted to Captain after Kirk leaves and Picard comes in as the new First Officer along with some other new crew changes. In this new scenario some of TNG characters mightn't make the transition. Unless the Klingons have been at least allied with the a Federation then you couldn't really reintroduce Work as part of the a Enterprise crew. You could have Worf as someone they encounter. Riker, Laforge, Crusher and Yar could be reintroduced as new crew. Wesley and Troi are two characters I would never care to see again.
So the ship's command history over the course of the series (assuming you could keep it running long enough) would be: Pike, Kirk, Uhura, Picard.
Well, it's just a thought. We are allowed to daydream.
I get what you're saying as far as realistic time frames for events and showing the passage of time. However this an endemic problem in all television shows. All of them, regardless of their other attributes and whether they're episodic or serialized, are in the "adventure of the week" format. If we think about it, almost all of the characters on TV live unbelievably eventful lives that no real person could ever live (even within the parameters of the fictional universe). The audience rarely notices unless this gets too out of hand and starts to erode the suspension of disbelief.
As for major changes in the cast, this is being done more often today. "The Walking Dead" is an example where main characters are killed off and others leave the frame of the narrative and then return later. However there are limits, most people (not me necessarily) need at least a few core characters to hang onto.