This question (or a close variation of it) was discussed in detail in:
Dawson, J. (2006). Why do mathematicians re-prove theorems?
Philosophia Mathematica (III) 14 (3), 269-286.
Dawson's article is itself a kind of sequel to an earlier (and well-known) paper of Yehuda Rav:
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica
(3) 7, 5-41.
In that earlier paper, Rav argued (p. 6) that
the essence of mathematics resides in inventing methods, tools,
strategies and concepts for solving problems.... Proofs, I maintain, are
the heart of mathematics, the royal road to creating analytic tools
and catalysing growth
and that in many cases (Rav, I think, would say all cases) the techniques and insights generated by a proof are more important than the result itself. In fact Rav refers to proofs (not theorems, but proofs) as "the site and source of mathematical knowledge" (p. 12).
In the subsequent paper by Dawson, this thesis is picked up and further developed. Dawson spends a fair amount of time examining the question "What makes two proofs different?" This turns out to be a fairly non-trivial question; for example, there may be structural differences, different strategies or techniques (e.g. one proof might use induction, another might not), and so forth. Dawson provides 8 different reasons why mathematicians might re-prove theorems:
- To remedy perceived gaps or deficiencies in earlier arguments;
- To employ reasoning that is simpler, or more perspicuous, than earlier proofs;
- To demonstrate the power of different methodologies;
- To provide a rational reconstruction (or justification) of historical practices;
- To extend a result, or to generalize it to other contexts;
- To discover a new route;
- Concern for methodological purity;
- To provide something analogous to the role of confirmation in the experimental sciences.
Dawson elaborates on each of those eight reasons in some detail, providing historical examples of how certain re-proofs play those roles.
With respect to your question "Is it worthy to spend time on doing research in finding other proof(s) for already proven theorems... rather than focus on unsolved problems or at least extending the edges of mathematics?", Dawson writes (p. 269):
Today, new proofs of old theorems continue to appear regularly and to
enrich mathematics. Indeed, in 1950 a Fields Medal was awarded to
Atle Selberg, in part for his elementary proof of the prime-number
So, yes, there is value in finding a new proof of an old theorem. Indeed, a Google Scholar search for papers with the phrase "a new proof of" in the title finds many examples of the genre.