Sounds like another one of those cool ideas which went all the way to prototype stage, and the bottom fell out at the last second. Except, the bottom didn't fall out; this system was actually put into use in real homes, 10,000 of them to be exact. This pilot program was considered successful, but legislators raised concern about the program encouraging children to gamble. After all, the Nintendo was largely considered to be a children's toy. The program was eventually scrapped for this reason, and was one of the last nails in the coffin for Nintendo's long-standing efforts at an online network for the NES.
But even if underage play wasn't an issue, this system was probably destined to fail anyway. The NES was about to be superceded by the Super NES and the Sega Genesis, among other systems, which would have required new hardware. And most importantly, the biggest opponent to home lottery would probably have come into play at some point, the Federal Government. The Feds have often used the 1961 Wire Act to shut down any kind of online wagering scheme, even legitimate ones. Even if underage play and the next generation of home game consoles didn't put an end to this program, Washington probably would have, even though it was restricted to within Minnesota.
As far as I know, this would be the last time a state lottery allowed players to wager from home. That is, until March of this year; when Illinois began online sales of Mega Millions and Lotto.