Start all the holes with the center spurred bit, drilling to about 1/2 depth. Then swap bits and finish with the flat bottomed one. There should be more than enough depth to get the second hole lined up, and it saves mutilating multiple bits just to drill multiple holes.
I'm boring 7/8" cold roll steel to get a flat bottom at 2.250" depth- .563 in diameter. What I've been starting with is peck drilling an 11mm hole with a carbide bit I have an abundance of, then following with a 1/2" x 4" long 4 flute carbide center cutting end mill to get a flat bottom, then using a boring bar to finish.
Likely any flat bottom drill or the like will tend to walk about at stating on a drill press. putting a flat bottom to an existing hole may be home made with putting a flat to a std drill or bought..An end mill use on a mill can start and finish a flat bottom hole.
The bottom of the hole will remain perpendicular to the long axis of the bit's shank, no matter whether the hole is being drilled squarely to the face of the board, or at an angle. For dowel holes, having smooth sides and a flat bottom give an ideal amount of mating surfaces for gluing the dowel into the hole.