I agree completely, and this is how I always envisioned it. Including the ship being more like the Conestoga.
Space is dark, cold, and foreboding. That's what TOS implied, especially about the pre-TOS period. The Constitution-class were the first multi-role starships capable of comfortably and quickly exploring the frontier.
But that was not the case pre-TOS.
If we go closer to the beginning, it's more like what happened to the S.S. Valiant.
The Valiant, one of the first warp ships humanity sends into the stars, gets off-course. 'Magnetic space storm.' A whole misadventure ensues. Ship is lost for 200 years.
That's what interstellar exploration was like at the beginning. There's a wonderful animation out there called "Death of the Valiant." That would be part of the vibe for an "Enterprise" series.
So the technology is really clunky and uncertain. Just going to warp is a big deal. Especially in early seasons. There would be a countdown and everything. People fastening seatbelts or strapping into crash harnesses. Inertial dampeners aren't perfectly tuned.
No transporters. Shuttles only. The ship can land if it has to.
If the NX Enterprise is an experimental ship, then it should look and act like one. Nothing is 'routine' about a mission to another star system. Including the smallest practices. Many interstellar 'missions' were one-way. Either by design, or by accident.
It would also be more 'political,' in a way that would build up to your proposed second series.
For example, I recently wrote [in another thread]:
I feel that a better "Enterprise" series would have explored more the contradictions between the exploratory and military aspects, being embodied directly in agencies such as UESPA and Earth Starfleet.
For example, Enterprise [let's face it, a different ship really] would have been launched by UESPA to follow up on its initial probes and colonization missions. In a more sober series, a "mission" could take half a season, and "Enterprise" would be part of a flotilla. It doesn't have the capabilities to do all the jobs.
Just following up on one colony could take like, six episodes. There are fewer colonies to explore in that era, more intrigue, and it takes longer to travel between them.
As things get more complicated, with the introduction of Klingons, Romulans, etc..., Archer's UESPA mission would start to really conflict with Starfleet. Who want a more defensive approach. And part of Archer's job would involve navigating these bureaucracies and the conflicts between them. UESPA funding is always in danger of being pulled, as politicians squabble back on Earth. [He'd probably need a political / diplomatic attache even more in this revised series.]
And because Earth is still recovering from world wars and the Eugenic War, the remnants of nation-state federations that exist on United Earth are jockeying for position, prestige, contracts, trying to push Enterprise in this direction or that. Altering it's mission. Pacific 201 explored this a little.]
[After posting that, I wrote some further notes]:
"Enterprise" is on a tight-leash. It is constantly reporting back home, whenever communications allow. [Often, they don't.]
The ship represents a massive investment on the part of UESPA, and possibly Earth militaries. The whole United Earth, which supercedes the UN, is breathing down their necks.
The whole mission plan was probably created in detailed form by a committee back on Earth, with different paths to follow depending on circumstances. Which would quickly come into contact with reality, and have to be further revised. Archer would be struggling against Earth bureaucracies over the mission plan. [Very limited time and resources to make decisions about.]
So Archer is not just a dashing cruiser captain exploring the far frontier. He has to become an Earth politician, of sorts. At least navigate it. He didn't expect it to such a degree when he accepted the commission. [The whole structure of managing a 'starship' is very new and shaky at this point.]
The ship wouldn't be called "Enterprise," because that introduces problems. It would perhaps be called "Intrepid," because our daring crew is intrepid. Or one of the TOS Constitution names, or named after the brave Valiant.
The "Intrepid" could even be the S.S. Emmette of the Ganges-class. A very rudimentary ship. But if it is storied enough, it could eventually become the delta symbol that is used on Enterprise, and eventually Starfleet. It could receive upgrades, such as a secondary-hull.
Honestly, the Conestoga would have been great, if going for a more rudimentary approach.
Maybe the Intrepid's first mission could be rescuing the XCV-330 S.S. Enterprise from some kind of disaster. That would be a nice tongue-in-cheek tie-in to TMP, and showing that this is not the settled Federation of TOS and beyond.
When things get closer to the Romulan war, there could be a need to produce a vast number of ships. Hence the Daedalus. Which could be the first "Starfleet" vessel. That would explain a lot.