According to the above principles, crown lengthening should be chosen, but if only the gum of this tooth is removed, the doctor will consider removing 3mm (you should be mistaken, 3Cm is too big), and the future restoration (tooth) will be too long and ugly. And if you are sure it is 3mm, then the cross section of the root must be lower than the top of the tooth, but your root is long enough, and the doctor is also considering keeping the residual root and carrying out experimental treatment. Generally, if the cross section is lower than the top of the tooth, it is necessary to consider tooth extraction. Crown lengthening is not an indication.
The main consideration of restoration is support, that is, your roots are long enough, so it is more reasonable to consider making a material with less irritation in the future.