There are a lot of factors which determine the ability to drill a concentric hole on a lathe and the chuck is just one. Are you using a center drill or to locate the center of the work or are you just plunging in with the drill. A center drill will act as a boring tool because of its rigidity and tend to correct any misalignment, providing a true center. Done correctly, the drill will center on the center drill hole. When you start the hole, do you notice any wobble in the drill bit. Allow the drill time for it to center. A hole started off center will only get worse as you drill deeper.
Even though you centered the tailstock with the alignment tool, that just means that it is good for that particular location of the tailstock and quill extension. A twist in the bed will change that as the tailstock is moved and if the tailstock isn't perfectly parallel with the spindle axis, both vertically and horizontally, any movement of the quill or extending the tool tip out from the tailstock as would be done by advancing the quill or adding a chuck and drill to the mix will change the alignment.
Aside from those considerations, it is possible to drill a hole off center even with perfect alignment. If a drill is pushed in without clearing chips,, commonly known as peck drilling, the chips can force the drill to one side. Once the drill starts drilling off center, it will only get worse as you continue. Even under the best of conditions, good practice dictates that hole depth be limited to 10X the drill diameter to control runout.
There are workarounds to drilling true holes on the lathe. One of the best is to drill a pilot hole and follow through with a boring bar. Another way that works fairly well is to drill undersized connecting holes from each end of the workpiece and follow with a slightly larger drill from one side. The initial hole will tend to guide the drill so it is centered either end an this will tend to center the drill in the middle of the workpiece. I tend to do this in three or more stages and the hole will generally be straight enough that a gage pin will slide through. Not as precise as boring the hole but it will work for less demanding applications.