Tooth extraction vs (imcomplete) Root Canal. Should I insist on extraction?

Approximately a year ago when of my lower molars got cracked. In itself it wasn't causing me day to day pain but would occasionally hurt if I bit down on something hard. Saw my dentist and he recommend having a crown fitted (which would have been my first one). So over the course of 3 months I attended four times for root canal prep work to be done.

After my final RC treatment I was due to attend to have the crown fitted but in the intervening period I found I was still occasioanlly getting pain in (what was left) of the tooth, which should not have been possible if the RC had been done fully. He was as puzzled as me as all the information indicated the RC was completed fully. Due to the pain he decided not to fit the crown and to leave the tooth as it was.

Since then I seem to be semi regularly getting infections in the tooth (each seemingly worse than the last). Over this weekend I've been in agonising pain to the point where I've had to take a day off work. Saw the dentist today and he put me on yet another course of antibiotics. I really did not want to lose the tooth and I've made a concerted effort to save it but today I went there wanting to get it pulled. However he was reluctant to do this.

Should I cut my losses insist this happens (as Christmas is coming and knowing my luck I'll probably get an infection over the holiday period). I'm also sick and tired of constantly popping pills and taking ever more powerful pain medication.

The RC work that was done did result in post and core work (where pins and cement was introduced into to gum to support the crown). Is it because of the presence of these that they're reluctant to extract? Or does it not matter?

Sorry for the overload of info but I'm going back on Friday and wanted some ammo to make my case.