This is actually pretty common in real life, with ships being launched at times as little more than an empty shell. Their main systems are then installed during the fitting out phase, and then there would be a period of shakedown cruises before it's commissioned and put into service. A better question would be why were there no other starships at all around Earth?
Maybe that was the only time Kirk and the other VI Ps would be available for the christening. As they approached their deadline they realized they wouldn't have everything ready, but enough for a quick trip, and so they went ahead with the launch rather than give up on having Kirk aboard. Still rather negligent.
It's also pretty clearly a press run - Harriman specifies that they're going not much further than Pluto and then back to spacedock. This is NOT a proper shakedown, it's just something done for the sake of the press or something. The Enterprise-E is said to have spent a full YEAR in shakedown status (though that length could have just been because it was one of the first active duty vessels of her class, unlike the Excelsior-class Enterprise-B), while this was a jaunt around the solar system and back to dock. It could be that there was some event or another that got Starfleet thinking 'hey, we've got a new Enterprise near completion, wouldn't it be neat to do some kind of celebratory press circuit or some such?' The Enterprise was engaged in what Harriman called 'a quick run around the block,' which means they were remaining close to major Starfleet installations - it's canon that there are Federation facilities at least out to Saturn's moons, so, given that it's humanity's star system, reaching out to Neptune or even Pluto and other objects in the Kuiper Belt with facilities is not out of the question. Under normal circumstances, if there were any injuries, if they weren't instantly fatal, they could probably make a quick pull over to one of them without much difficulty. Given the short trip they were expecting, it wasn't likely they would encounter anything too terribly dangerous, that they weren't expecting to engage in anything that would result in injuries a standard medkit couldn't handle.
The Nexus introduces a plethora of logic problems:
Why didn't Picard go further back in time to when Soran first arrived, so he could stop the whole thing?
Presumably he didn't want to introduce new problems into the time stream by going too far back, so he picked a time only a few minutes before his entry into the Nexus.
Where did the original Picard go when his future counterpart came out of the Nexus?
Maybe the Nexus also adjusts the time line when it inserts someone where they weren't before, so the original Picard was removed from the time stream (into the Nexus?) when his future self appeared in the same time and place.
The first time Soran got to the Nexus he was on a ship. Why couldn't he do it again?
That ship also blew up. Approaching the ribbon in a ship is dangerous, and since there's no guarantee that Soran would get into the Nexus before his ship blew up he decided to use a planet instead.
So we've got the Galaxy-class Starship Enterprise, the pride of Starfleet, one of the most powerful starships in known space against a much smaller, more lightly armed, obsolete Klingon Bird-of-Prey. The Enterprise has 10 phaser arrays which are arranged so that there are at least two facing you from any angle, can fire several simultaneously, and it can fire 5 torpedoes at once. Given this, does it seem possible, even likely, that the destruction of the Enterprise-D could have been avoided had they simply gone for constant return fire with phasers and torpedoes and simply overloaded the Bird-of-Prey's shields before they did too much damage? We only see a single phaser shot before they start looking for a way to trick the enemy cloaking device.
I pointed out on TNG's wallbangers page that it's even worse when you consider the events that lead to the destruction of the first Constitution-class Enterprise. Still heavily damaged from Star Trek II: The Wrath of Khan, operating with a crew of six under heavy, jury-rigged automation, and lacking the ability to actually raise her shields, she still managed to give as good as she got from a Klingon Bird-of-Prey.
Blink and you'll miss it, but at the very end, when Picard's going through the wreckage of his ready room, he finds the top half of the Kurlan naiskos that his archeologist mentor Galen gave him in season 6's "The Chase." He then casually puts it down and leaves it behind. Now, "The Chase" established that the artifact was absolutely priceless (beyond which, it clearly had sentimental value: he was deeply touched by the gift even at the time, and then to top it off his cherished mentor was killed only a few days later). Why would he treat it so offhandedly, let alone abandon it altogether, especially after it survived the crash intact?
WMG: The artifact was so valuable that he had it stored safely somewhere else (on Earth? in a museum?) and simply had a replica made for his ready room (we saw it on display in a few episodes of the series following "The Chase").