Conversion of Uniform Circular Array (UCA) into a virtual Uniform Linear Array (ULA). Is it practical?

Given the other questions you have asked, it seems you are moving REALLY fast into modeling and doing some of the processing you want to do. You've moved from basic STFT questions, to simple DOA estimation, pulsed-Doppler processing, and now into subspace algorithms like ESPRIT. It also seems you've fallen for the machine learning hype, as many of these processing techniques (especially for DOA estimation) are already robust in their "classical" form.

To answer your question:

The reason they split up the array into multiple arrays is a property of the ESPRIT algorithm. The fact that they start from a UCA and break it down into "equivalent" ULAs is expected. You would need to do this regardless of what the main array looks like. If one would have started with a single main ULA, one would still have to split it into multiple ULAs as well.

Without reading the entire paper (pay-wall) I'm actually surprised the authors are going through all this effort to "convert" the UCA steering vectors to ULA steering vectors. Given any array geometry, the steering vectors can be calculated directly. If you know the array geometry, you can calculate the time of arrival for each element.

Given an array of $N$ elements in a Cartesian coordinate system, each element has a position

$$p_n = \begin{bmatrix}x_n \\ y_n \\ z_n\end{bmatrix}$$

You then define the unit vector according to your desired angle of arrival given $\text{az}$ and $\text{el}$:

$$\hat{r} = \begin{bmatrix}-cos(el)cos(az) \\ -cos(el)sin(az) \\ -sin(el)\end{bmatrix}$$

Where the entries are negative due to the direction of the incoming plane wave. The time of arrival at each sensor is then

$$\tau_n = \frac{\hat{r}^Tp_n}{c}$$

From here you can convert the time delay to a phase delay to define the steering vectors $a$. Once you have the steering vectors, you can continue to beamform in the classical sense or using other techniques.