I just ran some numbers. In my 20 team league, getting 11/20 pts in each category you need (I chose to use 11/20 pts since if we assume 20/20 pts then C's contribution is far less to the total accumulated so it will skew the numbers):
734 R, 179 HR, 717 RBI, 97 SB, .280 AVG
Looking at the 20th ranked catcher in terms of 5x5 value according to Rototimes the last two seasons:
2006 - Laird 46 R, 7 HR, 22 RBI, 3 SB, .296
2007 - Snyder 37 R, 13 HR, 47 RBI, 0 SB, 0.251
Obviously some owners carried more C's but as you go lower in the C rankings, the overall production continues to drop off from Laird and Snyder. I just used them as the baseline production.

"with Laird" shows what each of your other 8 players will need to produce in order to maintain league average (or 11/20 pts) in all the categories. "without Laird" shows what each of your other 8 players will need to produce if you didn't have him. Same applies for Snyder.
Two things:
1. In this league, the impact of a C doesn't seem that big, even AVG when looking at Snyder
2. Having an active C really only helps you in 3 categories. Outside of a handful of C's, they don't really run
So the idea that having a C to rack up countable stats is probably true in most cases. I'm just not sure that if you really ahve a strong offense that you must absolutely have a C in order to win. They don't seem to make a big difference either way.
"Think of how stupid the average person is, and realize half of them are stupider than that." ~George Carlin