First, a comment. Fireplaces are much better for storing fire than piles of wood, and don't let anyone try to tell you otherwise. A full pile of wood only burns for around an hour, whereas a full fireplace lasts around 3. Also, even when a fireplace is down to embers, you can get it started again just by adding branches, so it actually lasts even longer.
I have several questions for trouble shooting.
1.Is there smoke rising from your fireplace?
2.Do you have any objects near your fireplace that might be blocking your pathfinding?
3.When you hold your torch with the cursor and right click on the fire, does you character move toward the fireplace? (try backing a long ways away and testing by spam right clicking all over the fireplace while the torch is on your cursor)
When you're trying to light a fire, watch the bottom left corner for the "the tinder drill broke" message. Tinder drills can break at any time during the lighting process, and there is a brief window where the percentage goes to 100% before the fireplace is lit, so that might be a possibility. (although the closest I've ever seen is one breaking at 90%) Also, make sure that your phlegm is full when trying to start a fire, because if you run out during the lighting process, you will stop and fail at lighting automatically.
As far as improving your chances with a tinder drill, I think you're just meant to upgrade to Flint & Steel, since it never fails. (although Flint & Steel is too expensive for people starting out, since it takes and iron bar and only has 10 uses)