Semi-native Chinese speaker here. (‘Semi-’ because I started both English and Chinese as a baby, but my English is definitely better since it was my everyday language. Chinese was used for watching dramas and Chinese homework. I know more Chinese proverbs though.) To learn kanji with (Mandarin) readings, frankly, I was systematically taught stroke order and readings at the same time. Stroke order stopped being explicitly taught at the point when it was assumed we could just copy our teachers’ actions. (It wasn’t true for everyone though.) Even then, I often looked up stroke order on my own, because I found it helpful for learning new characters.
Your question, however, is about how to spend less time on writing. I think the real question is this: what do you want to achieve? The easiest way to spend less time on writing is to write less. For example, if your focus is learning readings, write a kanji out just to familiarise yourself with its shape, with a stroke order diagram in front of you, and repeatedly say its reading aloud. Treat it like learning the alphabet in English: say a letter’s name, write it, and say its name again. You don’t have to memorise its stroke order since ingraining its shape into your mind should be helpful as well. On the other hand, if you want to be able to write kanji without a computer’s help, and perhaps even dabble in calligraphy, then you should try to memorise the order of strokes used. In that case, you have no choice but to find a way to memorise it. Ultimately, your goals will be a deciding factor. Nonetheless, I’d like to assure you that, as long as you pay attention while writing and do so fairly often, stroke order will eventually begin to stick.
A few other thoughts:
Both true. In particular, you may eventually discover this general trend: block by block, kanji are written from top to bottom, and left to right. Stroke order is something whose utility becomes more obvious when writing quickly: look at some 行書(ぎょうしょ) calligraphy samples to see how the strokes link up. Only a few stroke orders allow that sort of flow. That’s why they’re ‘standard’.
Each person has his or her own way of memorising readings. Mine is to visualise the character while saying the reading. That’s why stroke order helps so much. Also, I sometimes feel as though I can ‘see’ the sound contained in a certain stroke or angle between strokes. All this comes with familiarity, I believe, and that’s why I write kanji: I see them writing themselves in my head when I hear them. Mnemonics are great too, but only if they make sense to you. Otherwise, you’re just giving yourself a huge, almost random sequence to remember that might be harder to remember than the kanji itself. I know that I remember kanji as a combination of component meanings or as a combination of sound components and meaning components. I probably wouldn’t manage otherwise. However, to each his or her own.
At the end of the day, you need to examine which methods seem to be most effective in helping you reach your goals. I’d say the ability to write kanji, or at least familiarity with common components, is what helps native speakers with faster recognition and comprehension, and hence, I feel writing is very useful and should be an integral part of learning a language that uses kanji (like Chinese or Japanese). It also allows you to express yourself in writing without a computer or phone. I thus believe that any amount of time seriously and attentively invested into learning kanji writing is time well spent, especially since you’ll get faster with time. However, that may not be what you are aiming for, and you may find WK’s visual mnemonic approach sufficient, so the choice is yours.