Going by what I've seen on this forum and the people I know personally I would say many more people apply for FLR/ILR by post than in person. Going by this thread it seems like it's difficult to get an appointment anyway.
You probably won't have to wait for a result if you apply in person (same day isn't guaranteed..) but it's definately more hassle especially if you don't live near a PEO - your nearest PEO is Glasgow, my nearest PEO is Croydon which for me would have been a 2 hour journey each way and therefore would have meant a day off work. Far more hassle than just nipping into the post office and sending it special delivery.