I have had 6 or 8 drill presses over the years. Of the newer models, only the 10"+ bench top versions have enough quill travel (delta, craftsman, wen) all my 8" bench models had less than 2.25" travel. BUT I have had several older (1940s, 50s, 60s) presses and ALL of the older deltas (homecraft), Craftsman and Walker turners had at least 2.75" or even more travel. So I would look for some old $25 machines. You might even get lucky and find one with a $200 German chuck on it or an original jacobs. My experience has been that newer DPs have better on/off switches, enclosed belts, lights, and less rust . But older machines have better metal, more accurate, can be adjusted better. As for parts with older machines, other than delta dp14 bearings I think all other parts are easily available and cheap.
I think lathes are more accurate for drilling straight But I hate the idea of using my main lathe for drilling, not only is every minute of use, wear and tear on bearings and motors, But I just hate the constant switching of the chucks etc. Another possibility for you would be to buy a second lathe ($100 Harbor Freight or $50 tube craftsman to use as a permanent horizontal drill press). Although it is worth mentioning that a DP setup for lathes (MT chuck, blank chuck) is probably at least $80 for those 2 pieces. Also you need to be creative with ways to avoid tear-out in the lathe chuck