As between the versions for the same mora, they get sorted with tertiary differences as: hiragana comes before katakana, small comes before regular-size, and for katakana regular width comes before halfwidth. There's also a "circled" set of the katakana that sort after the halfwidth ones.
So they're equivalent (or not) depending on how you're doing the collation/comparison.
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That said, culturally Japan seems like exactly the kind of place where, were they English-speakers, all the kids would absolutely be required to learn perfect cursive.
In practice, there are no less than 4 separate scripts that are used in Japanese: hiragana, katakana, kanji, and romaji, and some mix of all 4 can appear in the same sentence.
It's not so much analogous to cursive, which is a different "style" of writing the same "thing" – katakana and hiragana developed at different times for different groups and came to play different roles, and there are (usually) semantic implications to which are used.
But from what I understand Hirigana is used more for Japanese words, and Katakana more for loan words from other languages.
It actually leads to a nice shortcut for some words. If I’m reading Hirigana I’ll try to match that with words in Japanese that I know. However, if the word I’m looking at is Katakana, I’ll flip that off and start trying to match phonetically.
I assume with fluency this all becomes automatic, but I’m a ways off from that yet!
Katakana, hiragana, and kanji are all in active use--that's why they don't fall away.
Kanji are your primary word base. They are sort of like root words in English.
Hiragana often serves as kind of a marker--endings of certain words as well as phrase markers (particles). These are particularly important because Japanese does not normally break words with spaces.
Katakana often denotes a foreign phonetic word or foreign names. Login is particularly good example for this forum: ログイン (ro-gu-i-n).
Japanese speakers actively use the differences as cues when reading. Watch a native Japanese speaker try to puzzle out Japanese learning materials for non-native speakers. If everything is written in hiragana (not uncommon for beginning materials), native speakers often have to puzzle over things a bit before they work out what a sentence says. This is one of the reasons why you want to get to Kanji as fast as possible when learning Japanese--the differences in script are important for reading comprehension.
You can see all of these in play on the Asahi Shimbun webpage: https://www.asahi.com/
Actually when you look at the use of English in Japanese media, you’ll quickly notice a lot of unnatural-looking overuse of uppercase. That’s because to them it feels natural to use uppercase the same way they use katakana.