Separate sets of Han characters cannot be encoded in the 16-bit space, but they could have been easily encoded in the current 32-bit space.
Nevertheless, I have never found this to be a problem in practice, because I have always taken care to have good separate typefaces for Japanese, Traditional Chinese and Simplified Chinese.
In documents that I create or modify, I apply styles with the appropriate typeface.
The only possible problems are with Web pages, but the good browsers allow you to configure typefaces for each language and I always configure the correct typefaces.
If the Web page does not specify correctly the language, it might be displayed wrongly, but this is only one of the many stupid things that can be done by a Web page designer that can make that page look ugly when rendered on other computers.
To avoid such configuration work when you prefer better looking typefaces instead of some standard system defaults would require a standardization of how to notify the applications about the association between certain typefaces and languages, e.g. by some environment variables or by some standard locations for the font files, depending on language.
While there were no existing legacy encodings allowing to write Chinese and Japanese at the same time, so there was nothing to keep compatibility with.
Can't blame them much though, han unification is a huge mess and designed by someone who I can only posit to be entirely brainless. There aren't many characters that are affected, you aren't even saving any considerable amount of codepoints. It's just west-centricism and lack of knowledge on the subject.