You're absolutely right about this with regard to the National Hockey League. If you've finished the documentary you've no doubt seen Igor Larionov - he came over with Slava Fetisov in the first instance of Soviet players coming to North America. Larionov had been drafted in 1985 by the Vancouver Canucks, and he and Fetisov were itching to get to the NHL and their story is told in the back half of that film. Larionov was joined in Vancouver by line mate Vladimir Krutov.

You've no doubt now seen the drama that constituted breaking down that wall within the Soviet system. Sovintersport needed cash, however, and some wage garnishing his made it easier for them to let go of the first wave of players, and other players to make a much easier transition. Fetisov's story seemingly went through in the film without mention of Sergei Starikov's - who let Mark Johnson's third period goal go between his skates at Lake Placid - drafting by and joining with the Devils in that 1989 season.

Other examples that come to mind are Slava Kozlov - drafted in 1990 by Detroit, he had smooth sailing coming over in the aftermath of the fall of the Soviet Union. He still plays in the Russian top league - considered the second best in the world - today. Sergei Fedorov was drafted in 1989 and came over the next year.

Pavel Bure was drafted in 1989 and came over after the August Revolution of 1991 - the Soviet Union still existed, but was in its final months and hardly a shell of what it had been. Still, he had an interesting case. Bure was clearly one of the best talents in the world, and the Red Army did not want him to leave. For his part, Bure later said he didn't defect because of potential ramifications on his brother Valeri (eventually an NHLer himself, and the husband of the oldest sister from Full House). His rights were bought from the Soviet federation and Red Army straight up in a courtroom in Detroit, for $250,000. The Vancouver Canucks had offered $200,000, Bure offered another 50,000 to sweeten the pot (The Canucks ended up covering that for him). This occurred TWO MONTHS before the fall of the Soviet Union, so it was still not all that simple like it would be just a year later with Kozlov.

One thing to note about hockey and draft picks, is that the vast majority of draft picks do not go straight into the NHL, they go through a complicated system of educational institutions, national teams, junior teams, minor league, foreign top-tier and farm teams. It is still common for European draft picks to not come over for a year or more, if they even make the jump to North America at all. In many ways, the time scale is still similar.

Why were these players drafted with no promise that they'd ever be able to come and play? Simple - the hockey draft involves a bit of fishing for longshots, and the top Soviet players of their day made it abundantly clear all over the world that they had some of the elite talents in the game. It was worth a shot in case it all panned out (in the spirit of glasnost!) - some of these players are absolute Hall of Famers; and the Russian game remains unique and a producer of top talent to the present day.

Finally, we have to mention the Stastny brothers, who made it possible for Eastern Bloc players to dream of playing the NHL in the first place. In 1980, Czechoslovakian stars Anton and Peter Stastny were at a tournament in Austria. Anton had been drafted by the Quebec Nordiques the previous year, and so Peter called the main switchboard of the Nordiques from their hotel room in Austria to try and arrange a defection. Peter had his wife come from Bratislava, and on the last night of the tournament they managed to "get lost" from the team on a night out in Innsbruck. They snuck to Vienna, where the Canadian embassy waited with paperwork. Their brother Marian joined them in Quebec the next year. They were the true pioneers of Eastern Bloc athletes coming to the United States. Peter was the second highest scorer of the 1980s behind Wayne Gretzky.

EDIT: It's been pointed out that I've forgotten a few important examples. Here's a couple more for ya:

Václav Nedomanský was actually born under the yoke of Nazism, and one of the top Czechoslovakian players of the 60s and 70s. He defected in 1974, the first pro hockey player to do so. After defecting to Canada through Switzerland, he played nine years in the WHA and NHL before retiring in 1983.

Petr Nedvěd is another good example from Czechoslovakia. As a teen, he came to Calgary for a youth tournament in January of 1989, and with $20 in his pocket, walked into the police station to defect. He's never revealed who helped him out, and until communism fell in Czechoslovakia later that year, his parents would call him to implore him to come back. Nedved is an interesting example because he was a youth player at the time - by the time he was eligible to be drafted in 1990, the government of Czechoslovakia had already fallen.

Finally, one more Russian example - the great Alexander Mogilny. Drafted in 1988 by the Buffalo Sabres after winning a gold medal with the Soviet Union in Calgary, the team helped arrange his defection the next year. He'd been playing on a line with Bure and Fedorov for the national team.

The other post about the NHL explains a lot about the later years, but I'll just add some more details:

At the time, when NHL teams drafted European players, they held their rights (within the NHL) indefinitely. So throughout the 1980s teams would draft players from the Communist Bloc in late rounds with the hope that one day they might be able to bring them over. Thus someone like Vladislav Tretiak was selected 138th in the 1983 draft by the Montreal Canadiens; the Canadiens did negotiate with the Soviet authorities to bring him over, but they didn't want to lose their star goaltender and refused (he then retired in 1984 at the age of 32, which was somewhat young, and I've read some reports that it was because he was denied the ability to move to North America and so wanted to spite the Soviets for this).

Other players were selected in late rounds as well. As noted, both Igor Larionov and Viecheslav Fetisov came over in 1989. But they had been drafted yeas prior: Larionov was picked by the Canucks 214th in 1985, while Fetisov had been selected 201st in 1978 by the Canadiens, and then again in 1983 by the New Jersey Devils (the Canadiens gave up their rights to him). Dominik Hasek, a Czech goalie who is regarded as one of the best ever, was drafted late in 1983 as well, and didn't come to North America until 1990.