The main problem with an attached sinktip on a regular floating line is that you will be casting much more weight than the rod is designed for. A sinking leader/lead-head is like a very short version of a shooting head. Adding a 7' super-duper fast sink leader can turn your wf4f line into a 6wt sink tip real quick. You will be overloading the rod and may not be able to cast it as far.
On the other hand, you could use this sort of set-up to have the same sinking line work on two different rods. For example, I've got a 150 grain Teeny-type line that I cut back to prolly 135 grains to work on my 4wt. I've got a 7' rio sinking leader than I can add to it and have a sinking line that will easily load a 6wt, and maybe even a 7wt.
The main advantage of a full sink line (assuming it is density compensated) is that you will have a straighter connection from rod to fly and this should help you feel strikes and achieve stronger hook sets as well. This can be a huge benefit if you are fishing stillwaters and needing to get a fly down quite deep. Sinking lines also cut through the wind better, due to their reduced diameter.
Sink tip and Teeny type lines sem to me to be a much better chouce for fishing running water because they allow you to mend line, but others have different takes on that.