RG:
The minhag of Klal Yisroel has become to stay awake and learn the first night only. I recall from my days of youth those who chose to repeat the second night as well. I will not address the “halacha” angle of this. But there is a very practical side here, and explains simply why the minhag became to learn all night only the first night.
What would be the effect of staying up both nights? Here are some thoughts:
1. One may believe that he is better than someone that doesn’t stay up the second night. This is in direct opposition to the humility that one should have as a learner of Torah, and the message of Har Sinai which was chosen because of its humility.
2. The body clock is built in to the human condition. It explains jet lag and the challenges adjusting to a different schedule. The exceptional up all night routine of Shavuos is a blow to this body clock (in scientific terms – circadian rhythm). The adjustment to correct this is challenging, but not fatal. Some have a greater struggle, and some handle this easily. If one did this twice, the adjustment to get back to routine would be that much harder. Sleep experts would advise against it. The result would be having a huge fight to function normally during the waking daylight hours of the following days. I never asked him, but I suspect that Rav Elyashiv ZT”L preferred to maintain his regular routine and not inflict such an attack on the schedule.
3. Labeling those that follow the minhag of Klal Yisroel, that exists everywhere, and is adopted by the gedolim of all the recent generations as “meikil” is blasphemous. One can always find ways to be machmir, and one can choose this for himself. That will never make the one that does not adopt the chumra a meikil. I think such a statement is an affront to Klal Yisroel, and constitutes one that becomes difficult to forgive – it would require asking mechila from hundreds of thousands of people. The Ari Zal slept on Rosh Hashanah afternoon. Was he a “meikil”?
4. There is a concept that requires Klal Yisroel to follow the guidance of gedolim of their generation. To hunt for shittos is foolish, and demeaning to the Gedolim that lead our generation. The trend today of seeking these obscure “chumros” and believing that we are somehow holier if we expect to make them the new standard is quite sick. This entire track of thinking is not consistent with Torah value. If you are studious enough to look for these “chumros”, why not redirect your energy to recognize the guidance of Gedolei Yisroel and to seek ways to see Klal Yisroel as righteous, not “meikilim”.