The lever stopping with a firm bite with 2cm to the bar on an Elixir does not necessarily indicate you could do better. Firm means there is no significant amount of air between the master cylinder and the caliper. Those two things can't be true at the same time. Even if a seal leak was somehow pulling in air from outside when you bleed the lever, and even if that air was somehow winding up in the Taperbore "reservoir" rather than the syringe, the fact that it's firm when you squeeze once pad contact has occurred means it's not where it matters.
With most bleed procedures that involve pulling on the syringe, it's possible to draw air in past the seals if you pull too hard. Without seeing what's going on it's hard to say whether this is true in your case, but the fact that it's possible doesn't indicate that anything is wrong. The pulling you're looking to do to draw air out of the lever shouldn't need to be super aggressive. You're gently applying the vacuum and waiting. If you find it's truly impossible to find a level of force where air doesn't get drawn in from outside, that indicates a leak.
Avid/SRAM levers and brakes have an o-ring on the bleed port screw and at the syringe fitting. One thing that can cause issues with leaks at the bleed port generally is if anything other than exactly one intact o-ring is present when either the syringe or the cover screw is installed (none or two, for example). Sometimes they can get shuffled around in the bleed process. It's also possible to damage the o-ring if the syringe fitting or bleed port screw it's around is overtightened. If your bleed kit doesn't have the o-ring, you could borrow it from the screw by gently removing it with a pick.
Remember that a leak at the bleed port that draws air into the syringe when vacuumizing the lever doesn't mean in any way that you're getting that air into the lever. One of the key things to be looking at is after you've done your last pull and push on the syringe and you remove it to go and reinstall the bleed screw (in addition to whether the syringe is leaving behind zero o-rings and the bleed screw is coming in with one) is whether the fluid level in the lever is full such that the bleed screw is displacing some amount of fluid as you install it. If it is full at the open bleed port, especially if you can situate it to be the high point, there is simply very little chance there's still air hiding inside the lever. Where could it be?
If you do have a leak that's caused elsewhere from the bleed port, yes a lever service can probably fix it. That would usually be accompanied by fluid leakage or trouble holding a bleed, which on Avid brakes almost always will mean a squishy lever feel.