To use a well-known quote: "It's a bold strategy, Cotton, let's see if it pays off for em".
So colour me surprised, but I was under the impression you were collecting royalties for hosting games on this site, not paying them. It's good marketing for the companies to get people into the game online, who then go out and purchase the physical game.
The nag screen, for me, is problematic. It in no way inspires me to start paying a tax to avoid it. It does, however, give me pause about playing here. Now I get it, I'm a freeloader of sorts in that I don't pay for a membership here, but that's largely because most board games are free to play online. Given that the standard for board games online is open source/free to play, it's my guess that putting up a nag screen and irritating your player base with hold timers before one is allowed to play a game. My guess is that I'm like most people, and it would sooner inspire me to just play elsewhere rather than buy into a pay model.
I'm thinking you guys should go with a different revenue stream. If you were profitable by showing everyone their ELO's, that seemed a legitimate policy, then. Personally, I never cared about them, but if you're providing value to those who do in exchange for a fee, then you have a useful free market exchange.
Right now it's just sort of a time tax.