Why wouldn't my general dentist bother with my leaked composite filling on a molar and opted for a root canal?

At this point, from the symptoms that you are describing, the tooth has become sensitive and indicates the likelihood for a root canal. More than likely, the decay had worked its way under the filling and has reached the nerve that runs inside the tooth. This would explain the sensitivity. The pressure you are feeling from chewing is indicative of a degree of infection in the bone of the tooth. Simply replacing the filling may not suffice. A periapical x-ray should be taken and a treatment plan for the possibility of a root canal should be considered. 
Healthcare is not an exact science and sometimes, at its early stages, a symptom needs to be closely monitored before a definitive treatment is indicated. Whether or not you would like to consider seeing a different provider relies on the relationship and trust you have with them.

Composite fillings can occasionally have marginal leakage around them leading to secondary cavity under a filling. These are not the easiest to detect and sometimes by the time we do detect them, its already too late. 
If you are having pain in a filling that has worked for 15 years, its possible that you have secondary caries, reaching the nerve. The only solution in such a situation would be a root canal treatment. Your dentist would however be the best judge of the situation.

Hmm, all composite fillings eventually leak. They are predominantly made of acrylic plastic which eventually breaks down. The average composite filling lasts 8.9 years according to a study conducted by the American Dental Association. Unfortunately recurrent decay around a composite filling can be difficult to recognize. Ceramic fillings look just as good as composite but are far more durable and often last your lifetime, but initial costs are greater. So think about this. You saved about $1000 when you got the composite filling in compared to the ceramic one, but now you need a root canal and a crown which will cost around $2,500 and you will be left with a tooth that is not quite as strong as had you had that ceramic filling placed years ago. So in the long run where is the savings?