I agree; but in all fairness, the difficulty in finding volunteer work for an individual goes beyond the NPS, federal government, and even non-profits that specialize in volunteer work. You would think such organizations would have someone that specializes in the logistics involved in getting individual and small-group volunteers to work; but the truth is, that they seem to only care about large groups of volunteers. With that said, many parks and services have "outsourced" their civic volunteer functions to larger organizations. The Friends of the Smokies is probably one such group that you are familiar with; for they are the pseudo-official branch for volunteering in the Smokies.
As for the remark about the parks being overrun; well I don't see that being the case from personal experience. I have been to many different national parks during many different peak seasons; and the closest one I've ever found to being congested was Yellowstone on a Labor Day Weekend. Even then, it still was not that bad. Great Smoky Mountain National Park is, supposedly, the busiest park in the system; and I've found it to very capable of handling the volume in the recent times I've visited over the years; including Memorial weekend, Independence weekend, and Labor weekend. I've never been to Yosemite, but based on pictures I've seen, it looks like the volume there can be incredible at times. Then again, it is also one of the very few national parks that is both uniquely amazing and yet so close to major population centers.
My guess is that the people saying such things are the ones that would prefer to see a year-round solitude like they are used to in the off-season.