Why does Deep Q Network outputs multiple Q values?

I am learning Deep RL following this tutorial: https://medium.freecodecamp.org/an-introduction-to-deep-q-learning-lets-play-doom-54d02d8017d8

I understand everything but one detail:

This image shows the difference between a classic Q learning table and a DNN. It states that a Q table needs a state-action pair as input and outputs the corresponding Q value whereas a Deep Q network needs the state as feature input and outputs the Q value for each action that can be made in that state.

But shouldn't the state AND the action together be the input to the network and the network just outputs a single Q value? I think this was just a "clever" design choice. You can actually design a neural network (NN), to represent your Q function, which receives as input the state and an action and outputs the corresponding Q value. However, to obtain $$\max_aQ(s', a)$$ (which is a term of the update rule of the Q-learning algorithm) you would need a "forward pass" of this network for each possible action from $$s'$$. By having a NN that outputs the Q value for each possible action from a given $$s'$$, you will just need one forward pass of the NN to obtain $$\max_aQ(s', a)$$, that is, you pick the highest Q value among the outputs of your NN.
It was designed such that the final fully connected layer outputs $$Q^\pi(s,\cdot)$$ for all action values in a discrete set of actions — in this case, the various directions of the joystick and the fire button. This not only enables the best action, $$\text{argmax}_a Q^\pi(s, a)$$, to be chosen after a single forward pass of the network, but also allows the network to more easily encode action-independent knowledge in the lower, convolutional layers.