Personally I can't accept the theory that there are more or even an equal number of Mechs and forces in private hands as there are in Govt hands, and they just step aside when wars come to their planets. It flies in the face of all other canon evidence and noncanon reasoning just to force-fit an answer to the question of reduced army sizes.
@Garrand - subinfeudation at least down to the planetary leader level is quite obviously the model used in Battletech, with varying degrees of loyalty to the lord paramount. The House Handbooks have examples of minor houses being awarded fiefs of factories, mining operations, asteroid belts, etc and the titles given to them as well.
@TDC - You ask why can't these other landholders just step aside and ignore a summons to help? They can't as they would be ostracised and probably stripped of their holdings immediately once the crisis is over. When the banners are called, you come - that's the deal. In a feudal or neo-feudal setting, this would be explicitly spelled out in a contract. IRL, it is merely implied in what is called a "social contract"... but it is still present. A close parallel would be political parties - if you are a Senator or MP or key Party head, on essential issues you do not vote against your Party. Go against your lord and you are stripped of your title - as Benito Mussolini was.
There is a confusion here with linking "feudalism" and "nobility" with mental images of knights in armour, and dukes and barons and so on. Actually, feudalism is (1 of) the roots of modern law and society. We still use some of the terminology of feudalism today. The simplest example is rent - you, the landlord, owns a freehold apartment. You let out the apartment to a tenant, who pays you rent and possibly maintenance. You in your turn pay the Government quit-rent. Your tenant might sub-let a room to another person - that is closely related to subinfeudation.
What's missing in the modern contract is that your tenant doesn't owe you fealty, and is not obligated to come to your aid when you call your banners ::) nor in your turn are you contractually obligated to do knight-service* in person to the US Army and raise a minimum of ONE (1) M1A2 Abrams tank, ONE (1) F-15 Eagle, and ONE (1) Company of M-16 Riflemen from your tenants, funded by your incomes... that went out with the introduction of scutage meaning the Government uses your tax-money to buy, recruit and train soldiers, instead of you having to be in the US Army yourself, and training in your own time to operate said Abrams tank.
But that is where Battletech reverts back a little to feudalism. As a planetary governor you are a Baron and preside over your planet. Your sons are probably Viscounts and have independent incomes from factories, farmland and mines - think trust fund babies :D But part of your contract to House Davion, implicit or explicit, includes the requirement that you have to hold on to your planet for Davion. (Otherwise he'll find a smarter baron to do it.) So you set aside part of the budget to raise a militia - and because you're smart, you put money into raising household troops too. Additionally, you send your young Viscounts to NAMA Academy where they'll train as Mechwarriors but more importantly, hobnob with Davion's kids and the other Viscounts and Viscountesses, where they'll find favour and contacts and perhaps rich mates - bit like the Ivy League, or so I hear - and you can point to them and tell Davion "Myself and my kids are Mechwarriors with our own Mechs. I hear your Davion Brigade of Guards is short on both. Why not take them on, as a personal favour from me... and by the way, did I mention my planet is opening a new iridium mine and we need export contracts?" Hint hint, nudge nudge.
And to connect all this to the topic - if you're especially savvy, you'll send word to Outreach that you wouldn't mind devoting a little more spare change than usual to hire an extra lance of Mechs as a little surprise, cause something tells you you haven't been raided in a while...
*sadly we are merely called conscripts these days :D