Actually, my understanding is that new calendarists typically do not fast before communion and receive every Sunday. This is because the hierarchs have followed the thinking of Fr John Romanides, who thought anyone who attended liturgy and didn't receive was a heretic simply for not receiving. Certainly, every new calendarist church I've been to gives this impression: EVERYBODY lines up to receive, along with their make-up, tattoos, dreadlocks, sleeveless tops etc. Perhaps some isolated conservative pockets, like the Ephraimites, fast, but they are the exception, not the rule.
As for the Kirykites, their problem is not that they fast, but that their Metropolitan has forbidden spiritual fathers from exercising any discretion over how much fasting to require of their spiritual children. Everyone in his jurisdiction must keep a strict fast from the preceding Monday, without exception. This is an inappropriately strict interpretation of the rules, but such an aberration does not in any way prove that the pre-communion fast is itself an aberration.
My own thinking on this issue is that if you are not willing to fast every time you receive, then you are probably not worthy to receive frequently without special fasting. Even if you argue that there are no canons requiring a pre-communion fast, that is basically like arguing you shouldn't make the sign of the Cross, because there is no written requirement for it. The Panteleimonites actually seem to think they are following a stricter practice by NOT fasting and by receiving every Sunday. People shouldn't fool themselves into thinking that by not fasting, they are actually returning to a stricter earlier practice, as if not fasting were somehow a manifestation of strictness rather than economy. Clearly, they are not fasting because they are too weak to do so.
Anyway, at least the written Slavic typicon requires fasting from the previous Monday for those who wish to receive on Sunday. I imagine this rule is not usually enforced as strictly today, but again, clearly we are seeing a condescension to people's weakness, not a manifestation of super-correctness. What I see with all this anti-fasting stuff is excuses to get out of the inconvenience of actually having to make an EFFORT and PREPARE oneself spiritually AND bodily to receive the Body and Blood of our Lord. Oh dear perish the thought!
On objection that makes some sense to me is: why don't priests fast before Communion? One reason, I suppose, is that they HAVE to receive every Sunday when they liturgize as part of their priestly duties. So it would put an unreasonable burden on them if they had to fast for three days EVERY week, which would really mean they would only eat normally on Sundays, Mondays and Tuesdays. However, Stavros Markou said that priests and monks actually should fast for at least a day or a half a day before receiving (if he's reading this, I'd appreciate his input). If priests don't fast at all, perhaps their bishop gave them permission to do so, but again, this is a relaxation of an earlier stricter practice where, at least on the preceding Saturday, at lunch they would go without meat and perhaps cheese, and in the evening meal would go without oil. If priests really cannot fulfill this (e.g. because of health issues), I can see why their bishop would relax the rule, but if they are just unwilling to keep even this little extra fast every week, I wonder if they are really worthy of the priesthood. Priests are meant to live by stricter standards than laypeople in any case, so why not by going without meat three days, rather than two days a week?
As to the oil on Saturday thing, I imagine that depends on the piety of the individual. Most of what I've read and heard tells me: if you will receive on Sunday, then no oil on Thursday and Friday (you shouldn't eat oil on Friday anyway, of course). On Saturday, you can have oil at lunch, but not in the evening. I've heard things about people sprinkling a few drops of oil on their oil-less food in order not to break the canon against fasting on Saturday. Either that is simply pious behavior, and shouldn't be condemned, or else it is a superstitious interpretation of the canons against fasting on Saturday. As far as I know, it's not that you HAVE to eat oil on Saturday, but that you CAN if you want. The canons were enacted because some bishops, especially in the West, were actually compelling everybody to fast strictly every Saturday in preparation for liturgy on Sunday, but the Eastern bishops were adamant that Saturday was still a holy day (the Sabbath), and strict fasting was not required. Of course, during the four fasts SOME fasting is always required even on Saturday and Sunday; it's just less strict than on weekdays.
Even those luminaries, like St Nicodemus of Athos, who thought even laymen should commune every Sunday, also thought you should prepare by fasting. All those saints who complain about those who only receive two or three times a year speak of their 'sloth' and 'indifference'. If you don't have to fast at all, how is it slothful not to receive, or how is it especially virtuous to receive every Sunday? All you have to do is get in line; no big effort there. The older advocates of frequent communion clearly envisioned the preparation for communion as a STRUGGLE, and were chastising their fellow believers for not struggling hard enough to receive more frequently. By removing the struggle from receiving, the new advocates of frequent communion are acting in the complete opposite spirit of the older advocates.