This is a memory from a couple of decades ago, so I could be wrong.
The real reason comes down to legibility. Kanji written badly can be hard to read, and Katakana or Hiragana written improperly can throw you off as well (め、ぬ、ね、れ、わ、ろ、る、シ、ツ、ソ、ン、ク、タ、ヌ). Japanese teachers, just like teachers anywhere, try to teach good penmanship not just as a virtue, but because it makes their own lives easier to stick to the rules and to have their students stick to the rules. The language itself has also undergone some reforms since the Meiji era to standardize and regulate it, including the stroke order.
This is a little off topic, but since I don't often get occasion to bring this up, let me drop it off here that there's also an alternative sort order which is an archaic (mostly, there are places that still use it), but cool sorting order called いろは ("iroha")[1] which is a poem which uses each character of the syllabary, each kana in other words, exactly once.
Also, if you open TextEdit on a Mac in Rich Text mode and create a list from the toolbar, there's an option to "Show More" which allows you to use いろは and other Japanese sort orders as bullet points.
These dictionaries are all 'Learner's dictionaries', and I'm not sure if SKIP was ever used in a dictionary aimed at native speaker - although it is probably the best method for paper-based Kanji lookup.
The more traditional stroke-count based lookup, OP was referring to is probably the radical+stroke count lookup, which is of Chinese origin and considerably older[1]. Using that system, you would first find out the kanji's radical (部, basically just a "part" in Japanese), find the radical in the index, count the remaining number of strokes in the kanji and then find the kanji.
Both the radicals themselves and the kanji listed below each radical heading in the index are ordered by stroke count, which qualifies this system as stroke count ordering.
This system is unfortunately much more confusing than SKIP (especially for learners), since there is a good chunk of kanji where the radical is not apparent, some radicals have multiple variants - and of course you'd have to first learn all the 214 standard radicals in the first place.
But besides Kanji dictionaries, most Japanese dictionaries (and other lists of words or name) are started by Gojūon (五十音), the standard sound order. If it is a specific word or name you're after, it generally has a distinct reading and it can by sorted by that reading.
There's multiple methods of searching a modern Kanji dictionary, but, yes, this is a useful property of stroke order. The bigger issue is when you have characters with 8+ strokes you quickly become reliant on specific shapes that come as artifacts to the stroke direction and order for proper identification, messing this up makes things a pain to read.
You also cannot purely rely on an "alphabetical" sorting in a Kanji dictionary anyway, as every character may have two pronunciations and you need to be able to look it up without knowing either. Some do have indexes letting you look a character up if you already know the on-yomi or kun-yomi reading but not the other, though.
Japanese does have its alphabetical order. It's called gojyuon-jyun (or aiueo-jyun) [1], and most commonly used dictionaries use this order [2].
[1] https://ja.wikipedia.org/wiki/%E4%BA%94%E5%8D%81%E9%9F%B3%E9...
[2] http://octoba.jp/wp-content/uploads/2015/11/jp.ne_.biglobe.s...
Words that may contain kanji that naturally has pronunciations are sorted by pronunciation, yes, but individual kanji never even had an inclusive list and thus no logically defined linear order existed/exist.
Here's an (online) example:
On the right hand side of the above site, it shows the pronunciation (both on-yomi & kun-yomi) and the stroke count.