Why use regularization instead of decreasing the model
Regularization does decrease the capacity of the model in some sense, but as you already guessed, different capacity reductions result in models of different quality and are not interchangeable.

L1 can be interpreted as making the assumption that the influence of different factors (represented by neurons) on each other shouldn’t be assumed without significant support by data (i.e. the gain achieved by larger influence has to outweight the L1 loss associated with increased absolute value of the parameter that „connects“ them).

L2 does the same, but makes this dependent on the connection strength, i.e. very light connections basically need no support (and are therefore not driven further to exact zero) and very large connections are almost impossible.

Dropout can be interpreted as training a large amount of smaller networks and using the approximated average network for inference: „So training a neural network with dropout can be seen as training a collection of 2^n thinned networks with extensive weight sharing, where each thinned network gets trained very rarely, if at all.“ Dropout: A Simple Way to Prevent Neural Networks from Overfitting

All these methods make certain network parameter combinations highly improbable or even impossible to achieve for a given dataset, which otherwise could have been the result of the training. In this sense, the capacity of the model is reduced. But as one can imagine, some capacity reductions are more useful than others.