If anything, it's more important to go to a Russell Group if you're an international student. These universities, no matter what the detractors say about it being a 'marketing tactic', are world leading, with the vast majority of them being in the top 200, or even top 150 in the world. I'm planning on doing my Masters abroad in Sweden, after finishing my degree at York, and it's stated in no uncertain terms that they will consider the international recognition of the university when considering an applicant.
Furthermore, the government uses Russell Group enrolment numbers as a metric to compare UK schools.
These are undoubtably some of the best universities in the country, and the world. Although, this isn't an absolute. Reading isn't necessarily a bad university, it's still ranked well internationally. That's what's important - don't focus on domestic rankings, they're a bunch of nonsense that change a lot every year and don't matter. Reading's ranked like 200 in the world, above some RGs. Lancaster is another good university not in the RG, which is around the same world ranking as York, 130-150.
In short, yes, going to a Russell Group is important. It's not just something that's just talked about in the UK, the name and prestige is world renowned.