It also lets shows develop and then feed into BBC1, rightly or wrongly. Without BBC2 we'd probably not have Peaky Blinders and Line of Duty.
The more I think about it, it dates back to as early as 1983, when Play School moved to BBC One in the mornings, (coinciding with its overhaul), whereas before it was repeated in the afternoon strand, and after Play Away's axing, BBC Two was deprived of regular children's programming until the See-Saw strand moved to there in June 1987.
Have I Got News for You?, which premiered in its first decade on Two, this October, marks twenty years since new episodes began premiering on BBC One.
Yes, daytime is a bit weak with it being too reliant on simulcasting the news channel, but before that all it did was show CBeebies / CBBC most of the morning anyway, so it isn't much different. Afternoons were weakened when content went to BBC1 but I think they've kind of established what they do in the afternoons now so it doesn't come across as much as filler as it used too. Personally for me BBC2 has been stronger than BBC1 in primetime in recent years - but I don't think one would work without the other.
Even so, ITV's always been one step ahead of the BBC in daytime: they have their mix of chat shows: (This Morning and Loose Women) and have really made it up in the late afternoon for the loss of the CITV strand in 2007, with game shows like Tipping Point and The Chase: quintessential teatime viewing, (though the Beeb did respond with Pointless on BBC One in 2012).