I am trying not to get the oven dirty by using a deep baking tray (great thing) or ovenproof glass dish.

One of the best chemicals for cleaning an oven (yes, the black crusts, too) is a not too strong distilled-water-solution of potassium hydroxide. Take care not to get skin or eye contact (!). Use rubber gloves and protecting glasses. Do not leave solution in the oven for longer time because it can be slightly aggressive against enamel. Rinse with water afterwards.
Oven will look like new.

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