I think your rules interpretation is sound, but that you underestimated the work needed on the covenant.
Most covenants, I take it, will need their grogs for guard duties, both at the gates and to escort teamsters to the market town (or similar), and also to maintain and mend their own weapons and armour. It is easy to forget this, because there are no rules to calculate the number of men needed as standing force. The blanket rule of two free seasons per year is just a reasonable approximation, reflecting that there is always some work which needs doing, and the grogs are there to do it.
It should be possible, I think, for a prosperous covenant to assign their staff to dedicated training and/or free time, but that would then be a covenant decision, and it requires spare man power. Elite grogs with four free seasons per year does not sound plausible to me, but three free seasons might if the covenant can afford it. Personally I would stick to the two seasons/year until we have established thorough book keeping of covenant resources and the records clearly show spare manpower, erring on the side of default.
But the main point is. Covenfolk /have/ two free season per year. The covenant can decide to /give/ them more. A wealthy character does not depend on the covenant. He /has/ three free seasons, and even the fourth he manages himself to maintain his income.