I've had the same exact issue with similar Shimano brakes and Avid BB7.
I normally put "preload" in the actuator arm, as I learned before on linear pull rim brakes (a.k.a. V-brakes).
What I do is to fully screw in any barrel adjusters and back up a turn or two, then with one hand I move the actuator arm to the point where it engages the disc, then tighten the pinch bolt just enough to barely hold the cable relative to the arm. There should be no slack in the cable. At this point I release the arm and pinch the cable with my fingers just before the actuator arm, as to prevent it from sliding too much in the next step. Now I press the brake lever roughly half of its travel, trying to reach just the desired engagement point. (since the pinch bolt is barely tight, it will let the cable slide). Then, without releasing the cable, the other hand releases the brake lever and fully tightens the pinch bolt.
The fine adjustment of the engagement point is done with the barrel adjusters and the pad position adjuster bolt(s) if present.
I use a similar method for v-brakes since those do not have a hard stop for how much the return spring moves, i.e. the brake arms opens as much as the cable allows. When I installed disc brakes I just assumed the same thing as the actuator arm usually moves way more (require more cable travel) than the cable travel provided by the lever.
Which brings me to a very important note: MTB Mechanical disc brakes require the same cable travel as linear pull or v-brake. Levers designed for road brakes or a very old MTB style of brake known as "cantilever" brakes are not suitable as they offer much less cable travel and cannot be adjusted properly in most cases.
Road disc brakes only became popular in recent years (answer written in 2021), so now there are many options of disc brake calipers that are suitable for road bike brake levers, as they are designed to work together. There also exists road brake levers that are designed to work with MTB types of brake.
Even though it is possible to mix Road and MTB brake components, it is really not advisable, because they are designed with different cable pull and leverage ratios (different mechanical advantage).
Just to clarify: I do not think this would be the case for the O.P. since he mentions the only changed component is cable, everything else is "as from factory". But someone mixing a normal Road lever with MTB brakes will experience a similar issue.