it would make sense, since it was supposed to be his clone
Except that a clone is not
actually a copy or identical twin, despite the tendency of fiction to misrepresent it that way. Your clone would be your offspring, your child -- just a child born without a second parent being involved.
And dramatically, thematically, Shinzon was indeed written and played as an estranged surrogate son for Picard. The emotional core of the film is that father-son dynamic between Picard and Shinzon. Picard sees himself reflected in Shinzon but is disturbed by the path his heir has taken and tries to guide him into making more of his potential. But Shinzon resents existing in the shadow of his overachieving "father" and is thus rebelling against him violently. The generational difference between them, the young upstart versus the wise elder, is essential to the character dynamic of the film.
Not to mention that watching Stewart and Hardy perform scenes together and play off each other is much more interesting and energetic from an acting standpoint than one guy doing a bunch of split-screen shots would be. Actors do better when they have other actors to respond to, when they can have a real dialogue rather than just talking to an tennis ball on a stick where a special effect will be stuck in later.