Binary classification model with time series as variables
I think one of your statements is simply a misunderstanding of the relationship of data structure and analysis. When you say "Melting the data, due to the structure, obviously wouldn't work too." you seem to failing to understand that melting the data properly would require construction of a supplemental covariate that would encode the "column location" of the values. In longitudinal data this location (in the "wide" version of the data) would become the "time" variable in the "long" format. Regression and neural network methods can handle such a format.

I intended that my comment only be a suggestion for improving a search strategy,  since you had not described the task or the inputs in any great detail. So I'm "answering" with some links resulting from several variations on a search strategy along the lines of ("binary outcome" OR "signal detection" OR classification) AND ("logistic regression" OR "neural networks") AND longitudinal AND missing:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5401893/

http://www.stats.ox.ac.uk/~snijders/WrightLondon2009.pdf

http://www.drizopoulos.com/pdf/Slides/jsm_2015.pdf

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3834930/

https://dl.acm.org/citation.cfm?id=3167596

https://mlhc17mit.github.io/slides/lecture7.pdf

https://www.cs.cmu.edu/~epxing/Class/10708-17/project-reports/project8.pdf

https://arxiv.org/pdf/1802.04664.pdf