It’s buried right at the end of a rather biased article (the USCCB didn’t say anything; a committee chairman did) but this really jumped out at me:
Teachers’ unions could be “permanently crippled” by the decision, the journal Education Next reported, though the decision could provide an impetus for other changes.
A loss in teachers’ unions membership could result in a decline in revenues and ability to affect policy. The National Education Association has planned a 13 percent cut for its two-year budget, totaling about $50 million, with its estimated membership losses of 300,000 people, about 10 percent.
Means that yes, they were in fact using “agency fees” to lobby for political ends, not (as legally required) only for union services.