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This is most easily demonstrated with the tritone substitution from your example.
The idea here is that the 3 and 7 of the chords are shared, which are the primary notes driving the function of the chord.
The effect I just talked about while not working for functional harmony may fit in a piece you are composing.
Progression 3 in general is not a very good way to imply I as tonic because you are letting one of the more powerful concepts in functional harmony slip away which is the resolution of the leading tone.
5 of the TT is ^b6, so if you have ^6 in your melody, you will have a half-step dissonance that may cause issues.
Same with 2 of the TT being ^#2 (b3), which could conflict with ^2 or ^3, or 6 of TT being ^#6 (b7) and conflicting with ^6.
The 5 of the iii chord is ^7 (scale degree 7), a half-step below the tonic, so if the melody lands on the tonic during the iii chord, it can cause a lot of tension, possibly beyond what you may deem appropriate or would be considered appropriate for a given genre/style.
Similarly, you can look at the tritone substitution (which I'll call "TT" moving forward) and find that certain notes may cause issues like this.
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I always like to remind people that music theory is not so much a set of rules, as many are often taught, but an explanation of what is happening and a language to discuss it.