The main impetus for founding new colonies post-2750 would be having a place to hide.
Most planets in BT are extremely sparsely populated*, and what population there is normally crams itself into a handful of ultra-dense urban areas, leaving the rest of the planet to agrarian use or completely abandoned. There's very little need for anyone to have an entire planet to themselves unless they're doing secret military/scientific stuff, or are in opposition to the local ruling faction. Even war refugees would generally have little need to go beyond the Inner Sphere when there is plenty of room on settled worlds in the relatively safe core regions of the Successor States and major Periphery kingdoms--except in the case of people who are political pariahs.
Even the Clans, with their inhospitable homeworlds and huge rate of population growth, didn't feel the need to colonize beyond the Kerensky Cluster in any serious way.
There will always be a few people striking out into the unknown looking for solitude or fortune or what-have-you, but that happens very slowly and sparsely. The age of mass colonization waves is pretty much over.
Even if there was any impetus to go out and expand into the Deep Periphery, there'd be pretty much nowhere to go. One of the reasons the Inner Sphere has the shape it does, is it constitutes a "bubble" of space that's conducive to life-friendly worlds. Garden worlds are very rare in the BT universe (to the extent that much of the IS as we know it was only made habitable by Star League terraforming projects), and beyond the near Periphery they become even less frequent. Whether it's a lack of water or the common star types being inimical to life in those regions, the areas beyond the frontier are fairly inhospitable. Up until just before the dark ages the level of technology needed to go out and settle those places was lacking. There may be more areas like the Inner Sphere with hundreds or thousands of habitable systems, but they're so far away that any colonies in those regions would be totally on their own, without even regular communication with the rest of mankind.
*Somewhat off topic, but how the IS is supposed to be home to "trillions" of souls is beyond me. Most of the IS and near Periphery planets are settled to the tune of a few tens of millions, or even less. The planets that have over a billion people are mainly sector capitals or major industrial worlds. One trillion, for the entire Inner Sphere, might be reasonable.