approach for multi label text classification I want to make a classifier that will label each text in a corpus with the correct label(s).
I can go straight to ML using sklearn multi-label text classification, or even to DL using LSTM.
But is it not better to start simple, and first use a rule-based system. That will help me understand the problem, and also set a benchmark accuracy score. Then I can make my algorithm more sophisticated (ML, DL) gradually in ways that only help the precision and recall.
So, you are asking about how to develop this system / model, which can classify text. 
Yes, it is a great idea to instantiate a "baseline" or dummy model, which can be rule-based or randomly assigns a label to a certain piece of text.
From this dummy model, yes you can then use RNN/LSTMs that does multiple-inputs (e.g. words in text) to single output probability over classes as a more sophisticated model and yes you would then compare the validation and test accuracy, F1-score, etc. to see if that improvement to the model is warranted by the change in the model's functionality to classify the texts.