Can there be some cure for dental caries without removing the affected teeth?

Yes: There are many procedure to save a tooth with carious lesions: root canal therapy, posts, crowns. These procedures can be performed only if there is enough solid tooth structure present and no periodontal disease. If there is not enough good tooth to warrant the time and expense of the above procedures, the tooth is deemed hopeless and extracted. $$$ can be better spent on an implant in that area.

Saving teeth: That's what dentists go to dental school for. To learn how to save teeth. From simple fillings to more involved endodontic, periodontal and restorative procedures. It's amazing what can be done nowadays. But we need our patient's cooperation in order to restore their teeth. At some point however, if the patient doesn't care or won't cooperate, or the tooth is hopeless, we can replace them.