My goodness so many people are missing the point of this thread.
The OP has mentioned a houserule to deal with his specific group at a high level, this is not a permanent change to all of Dnd-hood. So lets shove away all of the comments about concentration, and 3 casters being powerful and xyz, and just get back to the basics at hand.
The first question the OP implied, do I think the addition of more lower level spellslots will compensate for the loss of the higher level ones? That answer to that is no. In my experience (I have run one campaign that ended at 20th level in 5e), casters have enough of the lower level magics, especially when rituals are factored in, and those magics are not the equal of the higher level powers. So this houserule is a nerf to high level casters. Whether your group believes casters need a nerf at high level is completely up to your group, but I would say this is one.
Then the question comes down to...is the problem your trying to solve preventing the "nova" or the "everyday nova". For example, one houserule I've been tinkering with is the notion that high level magic slots are per adventure instead of per day. But they would keep their slots as is...or maybe even have more slots. So this means that a caster could throw down on a given day when they need to....but once that high magic is blown, its blown for a long time. Part of my desire for this is just to change the world issue of "why don't high level casters just run everything". If high level magics are literally moments of story, powers that a wizard may not recover for years (at least for npcs that don't adventure)....then it makes them a lot "cooler" and less spammable.