So as soon as the Enterprise had managed to get the barge out of the gravitational pull of the planet, they could have left it to coast to the Sun
But the barge was very heavy, and difficult to tow - and it takes a massive
amount of delta-vee to get objects that orbit a sun to fall into said sun.
A little nudge would have sent the barge coasting out of orbit and out of the star system all right. But to guide it to the sun, the Enterprise
would have needed to impart a lot of speed change / acceleration on it, and clearly this was not an easy or quickly achievable task.
The big question (after we choose to ignore the question of why go for the sun in the first place) is why they headed for the asteroid belt first. They could have spent the acceleration time struggling to force the barge out of the plane of the ecliptic, which is quite a bit easier than forcing objects into suns; the barge could then have been towed above or below the belt. But this assumes the belt lies nicely on the plane of the ecliptic; perhaps it's spherical, despite looks?
Wasn't it a case that they needed to tow it through the asteroid field because the barge's hull integrity had been earlier compromised and it would otherwise have fallen apart?
Pulling was the thing tearing the barge apart; not pulling would have been better. But the shields of the starship were needed to prevent further damage from asteroid impacts. No technique was mentioned that would have provided additional structural integrity to the barge, and indeed Star Trek in general lacks such a technique AFAIK.
Additionally, there's no reason to assume that Starbase 103 was outside of sublight range for the saucer.
Starbase 103 being within sublight range of Minos is utterly implausible. Minos was an enigmatic unknown, studied by probes only, and it took some time for Starfleet to send a second starship there to see what had happened to the first. Plus, if the starbase were anywhere near, LaForge should have summoned help from there - any runabout would have done in boosting the ability to locate Picard and Crusher and to outwit the killer probes.
SB 103 was very probably the nearest one, but not so close that sublight commuting would have been an option.