The Armed Forces of the Federated Suns (along with every other House military) can call upon all sorts of units. They've got battlemechs owned by the national government itself (just like the US government owns all the F-22s and Nimitz class aircraft carriers). They've got regiments that are the personal property of whatever Davion is on the throne at the moment (we don't have that -- the US President doesn't have a thousand Abrams tanks that he paid for out of his own pocket and that he keeps when he leaves office). They've got mech regiments that the Davion on the throne has as a result of his other titles (i.e., Duke of New Avalon has X number of regiments supported by the tax dollars of the people of New Avalon). They've got mech regiments that other high ranking nobles provide to the AFFS in exchange for recognition of their titles. They've got mech regiments that other high ranking nobles provide, but are generally limited to guarding their own planets (they probably can't pull units on garrison duty in the Capellan March that belong to the Duke of New Syrtis and send them to invade the Draconis Combine). Then you've got smaller units that lesser nobles have contributed, and owner-operators who provide maybe a single mech. We don't know what the numerical breakdown is, but all of them exist in significant percentages.
This is the way it broke down to me-
First Prince- AFFS, as set by whatever treaties integrated the Terran, Capellan and Draconis Marches into the Fed Suns which were ruled by 'princes.'
Prince of the Crucis March- Crucis Lancers brigade, maybe the Ceti Hussars
Duke of New Avalon- Avalon Hussars brigade
whatever of New Avalon City & surrounding regions- Davion Brigade of Guards
Duke of the Capellan March- New Syrtis Fusiliers
Duke of the Draconis March- Robinson Rangers
To me, those are the personal formations though the AFFS would provide support.
Then you get the regional commands though technically they are not part of the Regulars.
Marshal of the Crucis March- Crucis March Militia (CrMM)
Marshal of the Capellan March- Capellan March Militia (CMM)
Marshal of the Draconis March- Draconis March Militia (DMM)
Then you get stuff like the 1FSAC Regiment that are a national independent formation and assigned to a March by command. The Marshal and the Duke/Duchess of the Marches are not always the same person.
The AFFS roster does not typically include the personal regiments though some have a history of being bought into national service-
1st Kestrel Grenadiers- originally the Cunningham family
New Ivaarson Chassuers- maybe Stephenson family
Chisholm Raiders- planetary militia brought into national service
It does leave those niches where a noble has a battalion or company for their contribution to the national force- though with the naming conventions of the 3SW era, "Kingston's Killers" while otherwise known as the Marlette CrMM's 2nd Company, 3rd Battalion are the contribution of Viscount Danny Kingston of Yokelville on Marlette along with his retainers.
Raellus, for your last statement . . . yes, the Lyran Social General is a factor though it happens in all inherited titles. Then again, Caleb S-H-S-Davion blew his timing against the Caps and then Dracs . . .
Problem is, you have a LOT more nobles than you do national service academies. We probably have some sort of ROTC analogue for universities & colleges never mentioned in source material- or ones that get a throw away line. To compare, here in the US we have Westpoint (Army), Anapolis (Navy & Marines) and Air Force Academy in Colorado Springs) . . . but you also have private or public (depends on timing) military schools. The Citadel, Virginia Military Institute, Virginia Polytech, and Texas A&M. FREX TAMU is the largest of six 'senior military colleges' as set by law- currently 45% of the Corp on average receives a commission. During WWII the Corp produced 20,229 individuals who served in combat, over 14k who were officers- more than Westpoint & Anapolis produced at the time. Much to some ringknockers' dismay, Aggies have a enviable combat and career record despite being 'outsiders' from the national service schools.
Then you throw in that most US schools have a ROTC program as another source of officers. Finally, OCS is NOT an academy education which AW is conflating. OCS graduates are joked about as being 9-week wonders, the program takes college grads and turns them into officers though they lack the development of the academy grads. This path is typically where with a college degree they enlist and in contract are promised OCS if performance norms are met. After ROTC they will still send officers to a branch school- it might actually BE OCS since before that they are not strictly part of the US military- to learn their field.