I don't personally do dental implants but I can't specifically answer the question.
But generally, if there is a possibility that there may be an infection in the area (such as a crack that could hide bacteria), it may be a good idea to wait until the jawbone heals and the infection is under control. Unfortunately bone heals very slowly so it may be months before an implant can be placed. There are specific situations in which implants may be placed sooner but there would have to be optimal conditions for it to have a good chance of success. Your dentist probably feels that waiting would give you the best chance of success.
I know someone who had a tooth extracted and had to wait 6 months for an implant.
I had my extracted and a bone graft done or implanted and have to wait 3 months for an implant screw to be inserted into the bone.
Sometimes longer might be better I don't know, I'm curious. As to why one dentist might say 6 months and one might say 3 months.
It may depend on what they see regarding how strong the bone is maybe, but like your dentist stated, you had multiple roots. Who is doing your implant, a specialist or general dentist?
Bone does have to heal when a tooth is extracted I do know that and a dentist should always make sure there is enough bone there before doing an implant is what I read on here.