They don't need to give them access to military equipment. They could just train the client troops and guarantee their protection. The Marians have been trying to advance and mass produce even during the Jihad.
Which is not how client states work . . . you give them your second rate military gear to foster a dependency on you for their protection. You train their military leaders to influence their thinking, along with sending in advisors and training missions to make them align closer to your standards and be friendly. So, no if the the Marians produced a CN9-H2 or something that upgraded their old Hs closer to something like the CN10-B in loadouts they would sell/transfer their more marginal CN9-Hs to the client states. The client gets some adequate defense forces, they are dependent on the Hegemony for replacement parts and efficient training, and the Legions would still be able to roll over that client if it came down to it in the future.
Fronc is made up of old Capellan, Taurian and Magistry settled worlds. The Star League had bases & settlements there since the Reunification War, and we have indications there were states their previously besides being the center of trade routes between the TC/FS/CC/MoC. In fact, they refurbished a old Star League (or Star League era) facility on Detroit but took investment from nearby states IIRC. Rim Collection was producing weapons only until 'recently' when IIRC they started producing a local primitive mech at a single facility. Randis? Has a single facility producing a primitive mech but they got the benefit of a influx of Jag refugees.
None of which stack up to the Hegemony producing a handful of mechs along with vehicles and a pair of BA. Heck, the ONLY other Periphery power that builds two different battle armor (and they are the same class!) is the Canopians, and THEY got to that point by CapCon assistance.
The Rim Collection trading with the Magistry of Canopus for that single reference is indicative of nothing. I had a package shipped to me from China for something I ordered last year. I am involved in trade with China . . . but I do not compare with Walmart, even if we both trade with China. I am not claiming the Hegemony is not trading with the Dracs or even civilian side of the Alliance for a distant trade route (in fact, I point to the equipment buying missions), but you cannot build a economy on such infrequent or long distance trade. If they have jumpships engaged in that trade it means the ship is spending years going to that point and coming back . . . during that time it is not plying local trade routes. Instead of enriching local economies and worlds, instead your docking collars would be used to move DS in the League, Republic and Combine to use the closer example. Additionally, part of the reason the Rim Collection could have been trading with the Canopians could simply be b/c they wanted/needed medical advancements they could by there as part of Emma Centrella's periphery uplifting efforts that they could not get from Lyran space.
The referenced section of 'expanded production' was in relation to the establishment of a BA production facility on Illyria- which just happens to now be giving Marauder or Ravage BA to Tamarind I guess, it might also be the MH's only BA facility- but was in response to the League's reformation. Everyone bought into Stone's peace initiatives to one level or another, the Marians might have barely followed along like the Clans (no matter what, a more militarized society) rather than buying in wholesale like the FedSuns. You may not like it, but it was the state of the setting going into 3132- so at best they might have expanded the production capability since 3070s but not actually run the facilities at full speed except when they were replacing their losses. They were able to build new facilities to replace the destroyed ones on Perdition in 3099 on Addhara. FM3145 actually claims the MH has not seen economic & military growth after the settlement of the Lothian question that has not been experienced since the nation's founding.
Finally, you seem to be implying the Hegemony was under a single leader with a single vision- they were not. Julius, Cassius (d 3106), Lucian (d 3128)and Ignatius. The FM specifically notes that Ignatius has had less aggressive policies against un-aligned neighboring systems. Besides having to rebuild the I Legion twice in roughly 20 years, the other Legios were also engaged in widespread fighting. FM3145 says that settling with Lothians have only just begun to recover from the constant strain.