This subject is one of a lot of interest to me, so I'll weigh in
Firstly, there is a 'perception' that Alex Kay-Jelski (BBC Sport Director) thinks that way of the Boat Race, according to The Telegraph. I'll just say that that particular organ would have reason to hope that that is true, that the BBC has political motivations etc.
The Boat Race was, at a time, a genuinely big sporting event in this country, but these days, no-one really cares. People of my father's generation talk about a time when people would call themselves a "light blue" or a "dark blue" based on who they want to win, but it's relevance has really dwindled. This is a forum for sports nerds, and even then not too many people would talk about the Boat Race.
This is probably a nail in the coffin for it, although I'm not familiar enough to say whether it was a final one. I think of Lakeside Darts which was struggling to keep up with the PDC offering already, but when the BBC pulled the plug on it and it moved to Channel 4 it was game over.
Let's be fair though. I don't know how much it really is worth it for the BBC to pay using licence money for the Boat Race. I think people who live nearby may watch but I never saw it as an important sporting event, certainly not one with partisan feeling, and mostly important because of legacy and history rather than current relevance. I think it probably is a bit elitist, although I do worry about too many traditions being removed because of this perception, such as the Eton v Harrow match at Lord's.
In general though, in a time where it has been faced with a hostile government and print media, the BBC's sporting offering is still quite good. Obviously still a lot of football but that is the main sport in the country and even with women's football I'd say a good Lionesses run has huge social relevance in this country. The Women's Rugby World Cup which they put a lot of effort into their coverage of showed they don't just rely on football but are trying to put a lot into women's sport. Both rugby codes get a fair offering from the BBC, as does tennis with Wimbledon still a big showpiece for the corporation, and they still focus on snooker. Cricket, golf, and motorsport has been cut back massively; gone are the days where an F1 race would be live on the BBC probably forever now. The Olympics coverage I think is very good, to be honest having 25 streams for a billion quid is unjustifiable and Discovery is a fair offering for those who care enough about the Olympics to pay for it, I think the free offering from the BBC will suit a general audience.
Their athletics coverage is still among the best in the world but it's a big disappointment that the Swimming Worlds have disappeared, in my view for no reason. People still know the names of top swimmers and it's one of their favourite sports at the Olympics and Commonwealths and the rights would have been cheap as chips with Eurovision getting a hold of them. I'm really surprised ITV, C4, or TNT didn't want them either. I'm not too sure about cycling disappearing, although I'm not the biggest cycling head so I'll defer to you on that. Rowing and triathlon I think still get a fair enough shake. But I don't think that Auntie is just obsessed with football, although their interest in women's sport may bite for people not bothered with it. We do have to accept that football does hog a lot of interest and then there is a 'second tier' of sports with a lot of fans (F1, Cricket, both Rugby codes, and at 'peak times' tennis, darts, boxing, snooker and athletics) with the others going after scraps. Ultimately the BBC has to give the people what they want