Is verification of blockchain computationally cheaper than recreating it?
Yes, far easier
How do you verify the blockchain integrity? Don't you also have to recalculate all the values to see whether they are valid?
The miner has to find a value for parts of the block they can choose a value for, such that a hash of the block has a certain number of ...
The reason that is often stated is that you would have to re-calculate all the headers with all the hash values of the whole chain which is practically un-doable.
It's not the mere recalculation of new headers: miners currently create about 8×10^19 block candidates every second. However, at current difficulty levels, it takes about 4.8×10^22 block ...