If you reduce the chamfer at the front of the reamer, ie to get closer to the bottom of the blind hole, you will need to use a flat tool of some sort , and of course will need to reduce the amount left in the hole for the reamer. When I blind ream holes for pistons to use one circlip, I have made a pre reaming tool with a 0.1mm corner chamfer, essentially sharp, from centre cutting 3 flute or 4 flute endmills, ground about 0.1 to 0.15mm smaller than the final reamed hole.
As for the reamers, I buy the spiral flute ones with un even spaced fluting. These make very round holes right down to the bottom of the hole. I stop the reamer about 0.05mm from the bottom of the hole. A lot of places make un even spaced reamers these days