Just to stress a few points already addressed in comments and answers:
Euler in his time has at the same time discovered many important facts and solutions to classical questions, advanced rigor and given gave examples of the power of the recently created methods (infinitesimal calculus), popularized the science of his day (notably books dedicated to a German Princess), written wrote some of the first textbooks in analysis (still pleasant reading today), given gave strength to the prussian and russian academy of science, courtised courtized by two of the most powerful powers of the day (the King of Prussia and the Czar of Russia) Russia), filled international academic journals, some of them he edited himself, with quality articles (in fact up to several decades after his death because of the sheer size of his output), fostered international cooperation, wrote in the most important languages of his day (latin, french, german, I think he also learned russian), published in applied science, was part of state scientific advisory commission, etc.
In fact Euler's work has been instrumental in progressively establishing the "rigor" some of us are so proud of.
So a better equivalent of his investigation of what we call now Zeta(2 n) and the Gamma function would be the solution of an outstanding problem problems by one of the most recognized mathematician of his day building on recent work by one of his even more famous and established mathematician, Bernoulli, who was his PhD advisor and whose several family members have established positions in the scientific community.
I think he would have no difficulty publishing it. And his work would be quickly read and commented upon by many other mathematicians.
Even if we imagine a Leonard Euler finding himself straight jacketed straight-jacketed by the mathematical discourse and style of the XXIst century, he would quickly pair up with another good mathematician to write scholarly articles, as Ramanujan and Hardy used to do at the beginning of the XXth in a mutually benefical couple.