They're not problems in following the New Calendar per se, which is the context of my rebuttal to ilyazhito.
Personally, I would prefer the Old Calendar, but only because the New Calendar was forced on the Church of Greece and Constantinople. The calendar issue caused divisions in the OCA, with the Mayfield parish and a few other parishes in PA leaving for ROCOR because the bishop of PA in the 1980s forced Pennsylvania parishes to change to the New Calendar. In addition, there are some liturgically awkward times (St. Peter's fast and Kyriopascha especially come to mind) when certain liturgical features that happen under the Old Calendar disappear or are rarer with the new calendar.You do recognize that we still follow the Old Calendar for the Paschalion? If we also celebrated Pascha on the New Calendar, then the problems you identify would not be problems.
But you don't, so they are problems.
OK, but I'm not sure it's a valid rebuttal. The fact is that you follow a mixed calendar which has several serious flaws, and this shows up the error in the argument that the traditional calendar is composed of two separable parts, the Menologion and the Paschalion, and that one part could be changed alone without causing damage to the whole. The various problems of the mixed calendar, e.g. the shortened Apostles' Fast, or the impossibility of Kyriopascha, are evidence that the reformers were wrong in their belief that the Church had only anathematized the Western Paschalion, and not the Western Menologion: the calendar is a seamless whole and, like Christ's garment, cannot be divided willy-nilly.
Contrariwise, if, for the sake of argument, we adopt the new calendarist position that the anathema against the Gregorian Paschalion alone is valid, we then have to admit that reforming the date of Pascha is out of the question, since it has been fixed for all time by the Church, so that our choices only concern the Menologion, i.e. whether we should keep the traditional Menologion or adopt the new one.
I imagine you'll probably come back with the canard that the "revised Julian" is not the Gregorian. Of course, if that's the case, why ever did the original reformers not adopt a revised Paschalion along with the revised Menologion? In fact, we know the reason is that they did not want to incur the anathema against the Gregorian Paschalion, which on their interpretation was valid, even if the anathema against the Gregorian Menologion was not valid. So they implicitly acknowledge that there is no meaningful distinction between the Gregorian and the New Calendar with respect to the Church's anathemas.