one of the biggest things for breaking in an engine is to find the factory settings in the manual and set the needles to factory settings. it is a good idea to count how many turns it was as it came, but it is not uncommon to find that the actual engine is not at factory settings out of the box.
break in settings are typically very very rich. and depending on other factors, can be too rich for it to get running easily.
Please keep in mind what i am saying below is in my experience, and is for interpretation, i highly advise to get as much input as possible and figure out what you may think is the best approach for what you ultimately do.
i have rarely had issues to start a new engine that has been pre-heated to 220F, but if so i was taught to first start with trying to richen the LSN a few hours up to 1/4 turn, and if that seamed not to help the situation, then lean the LSN a few hours up to 1/8 of a turn from factory settings and see if that helps it start.
as with RC there is always exceptions, and there are some cases where the LSN is a little too lean at factory settings (dynamite .19 mach 2 for example), for some reason the manual says to set the LSN at 2 turns, where it seemed 3 to 4 turns is a better starting point to at least get it to run for most. one of my biggest recommendations is to pre-heat engine for break in. it will let everything expand, wont require the engine to try to push through such a high mechanical pinch, and allow the fuel and air mixture to be warmer allowing it to fire easier, running that rich it on its own wont usually hold its heat up as high, so insulating the head is also important (with tin-foil, a sock, or something), or bringing it back in to heat the head again with either a hair dryer or heat gun to get it back up to temp, also another note is to always return the engine to bottom dead center, where the piston is at its lowest point as soon as you can after the engine turns off and you are going to let the engine cool. (always let the engine cool completely between tanks)
i highly advise the Adam drake break in method (link below)
as far as the engine leaning out slightly causing the engine to run better as the tank gets closer to empty can be caused by a few factors, gravity weight of the fuel and pressure characteristics of fuel versus air when under pressure.
after break in tuning ill start with the HSN, lean it out 1/8 turn or so and slowly work my way in from there. i like to take have it be a pretty rich performance tune setting, and be easier on it for the first few tanks up to a quart to allow the piston to fully seat into the sleeve. when it does you will notice a pretty drastic change for the better in its performance, then finish performance tune.