The way I read van Buuren's background on imputation, and this part specifically, is that for multiple imputation models the goal is to use as much information as you have in order to obtain the estimates required to complete any missing data. This is grounded on the idea that multiple imputation is founded on the missing at random assumption.
Consequently, overfitting the imputation models - i.e. making it more dependent on your data - is less of/not a problem: any available associations, small or large are used. If these associations are estimated with large error, it will be reflected in the variance across imputation sets and taken into account when properly pooling analysis results. On the other hand, if the estimation error of these 'overfitted' errors is relatively small, it may improve the estimates by reducing variance across imputation sets.
I feel this reasoning applies to multicollinearity as well. I assume van Buuren supports this, but I think the van Buuren statement you quote is more related to the practical implications of modelfitting under circumstances of high multicollinearity: your computer might start spewing pink smoke, while not being able to fit the model at all!
Even if so, I feel there are no obvious reasons to state multicollinearity is more of a problem than for example overfitting: it's just about using as much information as is available. The only difference is that in the case of multicollinearity you might want to complete this with it's just about using as much information as is available and your are able to fit.
So if the computer throws an error, you might have to 'dumb down' your model. However obtaining replacement values through a less multicollinearity sensitive model might also be an option.
Sidenote: if you have a set of perfectly multicollinear variables, they're probably identical, or completely dependent on one and other, and on or the other does not provide additional information for estimating replacement values. In that case removal of one of the variables is probably indicated.