I am curious to know the best way to extract the seasonality from the time series other than the seasonal indices way (i.e., for better accuracy in capturing the true seasonality).

There are several questions to unpack here about seasonality. (1) Is seasonality best approached as a multiplicative factor? (2) Is seasonality best approached through a fixed-size vector reflecting those factors? (hence the "profile") (3) How to compute the values of those vectors?

Concerning (2), the use of a fixed size vector to reflect the seasonality (like a 52-week vector) has some limitations. For example struggles to capture patterns like an early summer. More generally the vector approach does work too well when the seasonal pattern are shifting, not in amplitude, but in time. The literature provides more elaborate approaches like dynamic time warping (DTW). However, DTW is complicated to implement. Nowadays, most machine learning researcher have moved toward deep learning. However, I am on the fence on this. While DTW is complicated, it has the benefit of having a clear intent model-wise (important for whiteboxing).
https://en.wikipedia.org/wiki/Dynamic_time_warping

Finally (3), the best approach that Lokad has found to compute those vector values is differentiable programming. It does achieve either state of the art results or very close to start of the art with a tiny fraction of the problems (compute performance, blackbox, debuggability, stability) associated with alternative methods such as deep learning and gradient boosted trees. The method is detailed at:
https://tv.lokad.com/journal/2022/2/2/structured-predictive-modeling-for-supply-chain/

There are several questions to unpack here about seasonality. (1) Is seasonality best approached as a multiplicative factor? (2) Is seasonality best approached through a fixed-size vector reflecting those factors? (hence the "profile") (3) How to compute the values of those vectors?

Concerning (1), the result that Lokad has obtained at the M5 competition is a strong case for

seasonality as a multiplicativefactor:https://tv.lokad.com/journal/2022/1/5/no1-at-the-sku-level-in-the-m5-forecasting-competition/ The literature provides alternatives approaches (like additive factors); however, this don't seem to work nearly as well.

Concerning (2), the use of a fixed size vector to reflect the seasonality (like a 52-week vector) has some limitations. For example struggles to capture patterns like an early summer. More generally the vector approach does work too well when the seasonal pattern are shifting, not in amplitude, but in time. The literature provides more elaborate approaches like dynamic time warping (DTW). However, DTW is complicated to implement. Nowadays, most machine learning researcher have moved toward deep learning. However, I am on the fence on this. While DTW is complicated, it has the benefit of having a clear intent model-wise (important for whiteboxing).

https://en.wikipedia.org/wiki/Dynamic_time_warping

Finally (3), the best approach that Lokad has found to compute those vector values is differentiable programming. It does achieve either state of the art results or very close to start of the art with a tiny fraction of the problems (compute performance, blackbox, debuggability, stability) associated with alternative methods such as deep learning and gradient boosted trees. The method is detailed at:

https://tv.lokad.com/journal/2022/2/2/structured-predictive-modeling-for-supply-chain/

Hope it helps, Joannes