Imagery vs. Information
Try staring at the kanji 「湯」 for 5 minutes and tell me what it enabled you to "see" and/or "imagine". Did you have fun?
I did it before posting my answer and all I "saw" was hot water. I had no fun staring at the kanji. Kanji have specific meanings and that means there is not much room for imagination. 「湯」 means "hot water", but who does not know that a public bathhouse has hot water?
Now try staring at the hiragana 「ゆ」 for 5 minutes. What did you see? Maybe not much if you do not possess a large vocabulary in Japanese.
I started staring at 「ゆ」 and within 30 seconds, I was already in ゆ-topia with the nice images conjured by the kana.
I was soaking in the perfect-temperature ゆ very ゆったりと("relaxed") and ゆっくりと("leisurely"). I obtained 心のゆとり("ease of mind"). Thank ゆ so much, public bathhouse!
If we have three writing systems, we might as well take advantage of each of them? Who cares if that caused Japanese-learners confusion? English was never an English-learner-friendly language, either!
Above is just a personal opinion of a native Japanese speaker, ゆ know.