But maybe there is a simplier reason than adjusting your lathe.
We have probably all observed this lift even when we can assume the tail stock is aligned to the center line, the chuck is straight, and the drill bit is not bent. The drill point should touch at the center of rotation.
Flute cutting surfaces are not perfect. Consider a two fluted bit. So on very shallow cuts, i.e. at the beginning of the cut the longest cutting edge engages first and if the dept rate is slow then this is the only flute that actually cuts. This then would cause the drill bit to be pulled to the side asymmetrically. If both flute cutting edges are perfectly matched they touch at the same time and cut at the same rate and the side pulling would be equal. Then the force on the drill bit would be a symmetrically induced torque. If you have a weak or loose hold on the bit you will find that sometimes it is not a lift up, but a pull downward or towards a side. If the chuck mount is loose and the drill bit is long you will actually see wobble.
When the two sides of a cutter are not matched you will commonly see the spiraling cut waste coming from only one flute. When they are perfectly matched you will see both spirals coming off equal. However, if the cut is shallow (low feed rate) this is rare! The cutting edges are commonly not perfectly matched and this is another reason to know the depth of cut rate (feed rate). If only one flute is cutting it is common that the other side is just rubbing and generating heat and quickly getting duller. If the plunge rate is such that both flutes (even unequal flutes) are engaged there is less of the sideways pulling, heat, and the bit stays sharp longer.
Experiment to run. Choose a poor quality drill bit that has unequal length flutes (one does not cut). Put the bit in the tale stock, after facing the material to be drilled then, using a low RPM, start the cut and observe the pull. Now repeat the process after the drill bit has been rotated 45, 90, 180 etc degrees and observe which way the pull happens. (If you really want to see the effect use a long small diameter drill bit. It won't have to lift the tail stock, as it can easily deflect/bend.) The larger diameter, stiffness, of center drills are are largely why they are used. Also another reason to purchase good drill bits and to start the actual drilling with a short stiff bit.
For those of us that only have grinder to sharpen tools with, sharpening smaller drill bits (~<1/4") to have equal flutes is virtually impossible. A good machine to do this with is quite expensive ($K, not unreasonable) and one can purchase a lot of small bits for the machine cost! As they dull I move these bits from the machine shop to the wood working shop and then finally to the catch all where about all I use them for is drilling in plaster!
Trick: By the way, my 90+ year old house has really hard plaster and on the outside ways this plaster is placed directly on tile blocks. To hang picture is a challenge. If you start to drill into this stuff (masonry drill bits) you will find that the bit commonly walks around and locates to a low or soft spot. The trick I have found is to start out by running the drill backwards so that the drill bit just rubs creating the low spot where you want the hole. (You can do this with a worn down bit.) I guess you could call this grinding the starting spot!
Dave L.