I’m not sure it’s strictly true to say it “does not come across as too harsh or upsetting“ - the opposite warning should probably be applied. It’s a very strong and direct way to say no and for that reason it is nowhere near as commonly occurring as the English “no” in everyday speech; people tend to soften their negations and rejections by diluting them across several vague words. For example, if someone at work asks if you have time for a meeting later today “no, I can’t today, sorry” is a perfectly fine answer, where as 「いいえ、今日は無理です。ごめん。」would be quite surprisingly direct. You’re more likely to hear the far more indirect 「それはちょっと難しいですね。」
But 「それはちょっと難しいですね。」 means "It is difficult, isn't it?". I think 「いいえ、今日のありませんです。すみません。」is more direct. I'm native in Japanese and I'm sure this is right.
