Learn English – Does one use a hyphen in between ‘other’ and ‘hand’ when saying “on the other hand”

The short answer, I think, is that the em-dash makes it easier to read the sentence and thus more likely that the reader will understand your point and want to continue reading.

Readers today are inundated with far more to read than ever before, and any structure that forces the reader to do more work will be detrimental to their likelihood of reading further.

Grammatically, semicolons are used to connect two independent clauses without the use of a linking preposition. However, the type of connection is not stated outright; the reader is supposed to assume a reason for not separating the clauses into sentences. (In the last sentence, the implied connection between clauses could be expressed by the word therefore.)

When a writer uses an em-dash rather than a semicolon, there is white space around the connection. White space, particularly for visual learners, signals that there is some work to be done by the reader to understand the connection between two things. It also allows metaphorical space for that to happen, since it forces the reader to pause and make the connection.

A semicolon doesn't offer that same amount of white space. It is a subtle cue, and can slow the reader down if they don't see the period above the comma. (Anything that makes a reader pause and say "huh" will increase the likelihood that they will not continue to read.)

The hyphen is not required in the first pair. As you say, high is simply an adjective and performance is simply a noun which high modifies.

In the second pair, high is again an adjective and performance is a noun functioning as an adjective and together they modify computing. The question to ask is whether the absence of a hyphen between high and performance creates ambiguity. Is it possible that a reader might somehow think that computing is modified independently by both high and performance? Probably not. Nevertheless, as the late R L Trask wrote in ‘The Penguin Guide to Punctuation’:

The hyphen is . . . used in writing compound words which, without the hyphen, would be ambiguous, hard to read or overly long. Here, more than anywhere else in the whole field of punctuation, there is room for individual taste and judgement;

The absence of a hyphen might cause the reader a brief moment’s uncertainty over whether high and computing are to be taken separately or as a whole. If you think that might make the piece ‘hard to read’, then you would be well advised to use the hyphen.