Kalman filters vs. state observers

They are both different forms of the same thing. I will try to clarify this a bit without a lot of buzzwords, which many people seem to do.

In short: The Kalman filter is an optimal observer similar to how LQR is an optimal controller. Both together form an optimal state feedback controller.

A bit longer: You already noted that an observer is nothing more than a kind of controller controlling your estimate of a system towards the real system based on your sensor output, so just think of it as a controller. You can design a controller using frequency domain techniques giving you a transfer function with poles and zeros, using a state space representation and providing a control matrix yourself, or by solving a Riccati equation to find an optimal controller (LQR), requiring you to define cost matrices penalizing control effort and error. (And in many many other ways.)

The Kalman filter is just an optimal observer/estimator similar to how an LQR controller is an optimal controller. You can even design them using the same Riccati equation. (You just change your weighting matrices to the covariance matrices of process and observation noise and so on, but you are familiar with this.) The Kalman filter or Linear Quadratic Estimator (LQE) is a way of selecting the observer gains. You did this manually using pole placement to stabilize your error dynamics (see the controller analogy?). The Riccati equation gives you stabilizing gains by definition, but you lose control over where poles are placed exactly.

If you (or anyone else reading this) want to know more about this, the wikipedia page about Linear Quadratic Gaussian Control (LQG), which is nothing more than the combination of a Kalman Filter and a LQR controller might be a good starting point.