Why do RNNs usually have fewer hidden layers than CNNs?
CNNs and RNNs feature extraction methods:

CNNs tend to extract spatial features. Suppose, we have a total of 10 convolution layers stacked on top of each other. The kernel of the 1st layer will extract features from the input. This feature map is then used as an input for the next convolution layer which then again produces a feature map from its input feature map.

Likewise, features are extracted level-by-level from the input image. If the input is a small image of 32 * 32 pixels, then we will definitely require fewer convolution layers. A bigger image of 256 * 256 will have comparatively higher complexity of features.

RNNs are temporal feature extractors as they hold a memory of the past layer activations. They extract features like an NN, but RNNs remember the extracted features across timesteps. RNNs could also remember features extracted via convolution layers. Since they hold a kind-of memory, they persist in temporally/time features.



In case of electrocardiogram classification:

On the basis of the papers you read, it seems that,


  ECG data could be easily classified using temporal features with the help of RNNs. Temporal features are helping the model to classify the ECGs correctly. Hence, the usage of RNNs is less complex.


The CNNs are more complex because,


  The feature extraction methods used by CNNs lead to such features
  which are not powerful enough to uniquely recognize ECGs. Hence, the larger number of convolution layers is required to extract those minor features for better classification.


At last,


  A strong feature provides less complexity to the model whereas a
  weaker feature needs to be extracted with complex layers.




Is this because RNNs/LSTMs are harder to train if they are deeper (due to gradient vanishing problems) or because RNNs/LSTMs tend to overfit sequential data fast?

This could be taken as a thinking perspective. LSTM/RNNs are prone to overfitting in which one of the reasons could be vanishing gradient problem as mentioned by @Ismael EL ATIFI in the comments.

I thank @Ismael EL ATIFI for the corrections.