Why would they do that? Their relatives diverged from them and went southwest to invade Western Roman Empire some six hundred years ago, and there would be very little contact since.
I've been skimming Wikipedia on the subjects of Goths and Arianism. Arianism as such is an idea that has apparently re-emerged spontaneously over the centuries; it seems to me at a glance that many modern sects that can be classified as broadly agreeing with Arianus probably came to these doctrines by independent reasoning of their own, and in specifics they tend to disagree with specific judgements of the classical Arians.
Meanwhile, Swedish "
Gothicismus," or as English would put it, "Gothicism," is a modern romantic-patriotic construction.
It seems to me the evidence does support the supposition that modern Swedish territories that retain a "goth" root in their names do indeed have to do with being an ancient homeland of the ancestors of the historic Goths who did convert to Arianism, and therefore that modern Swedes do include in their ancestors a relationship with these Goths. Whether it is reasonable, as you have assumed, that a distinct Gothic people existed in the tenth and later centuries who are substantially the ancestral stock of the historic Goths, and themselves continuously identified as Goths, depends on details I don't know but I guess it is not unreasonable.
What is very dubious indeed is the suggestion that once the migratory branch of the expanding Goths who moved south across the Baltic lost a continuous connection with their ancestral home in Scandinavia, that any significant connection between them and their wandering cousins remained. Of course it is not impossible that even as the distance widened, with the great Goths of late Classical history appearing on the Roman horizons very far south and east, in Ukraine and nearby places in Romania, Moldova and Belarus, that knowing their old kinship individuals traveled back and forth and communicated news.
But if so, I don't see any sign, not OTL anyway, that this news significantly affected anything the northern stay-at-homes did. So, with mass conversion of southeast European Goths to Arianism due to the missionary work of Wulfias in the middle of the 4th century, there is no sign that anytime in the subsequent three or so centuries that Gothic/Vandal/Burgundian Arianism had any resonance or echo whatsoever in Scandinavia. By the end of the 7th century, the last of these Gothic Arians had been converted to either Roman Catholicism (in France and Hispania) or Eastern Orthodoxy (in Anatolia) and any remaining holdouts in North Africa would have been absorbed by the general conversion of that region to Islam a century later. So long before the Scandinavians generally began their stirring and expansion we call the "Viking Era," the whole tradition of Germanic Arianism had been extinguished.
For a plausible connection, we'd have to see some sign of Arian Christianity being established in the Gothic parts of Sweden/Denmark long before 700 CE, which is also long before the OTL Roman Catholic missionaries began making any inroads in the far north either. Since Orthodoxy in Russia is a phenomenon of conversion long after the Viking influence created Rus, also some centuries after the final last gasp of Gothic Arianism known to history, it would follow that Arian Christianity in ancient ancestral Goth-land would be by far the earliest intervention of Christianity in any form in Scandinavia. Obviously, given the difficulties Catholic missionaries faced, we'd expect some serious conflict and controversy in the northlands. Or if for some reason or other the Arian approach and doctrines went down smoothly among Gothic northerners, it would probably have the same chances of success among the non-Gothic Scandinavians too. Then the Viking Era, if not butterflied away, would not have been a matter of pagans raiding, pillaging and conquering, but rather (from a Catholic point of view just as bad or worse) Christian heretics doing so instead.
It would be completely unreasonable to have just the Goths convert to it, and participate in the Viking era alongside their pagan neighbors.
Clearly OTL any possible connection that might have led to north Gothic conversion failed to do so; it seems plausible enough that a few individuals may have converted but clearly this trend died out if it started at all. It might not take too huge a butterfly to make it happen anyway, but if it did the consequences would create major upheavals in history that could not possibly be overlooked. Either we wind up with Arian Scandinavia as a whole, or a civil war in which Gothic Arianism is again overwhelmed and suppressed.
For the Goths and Goths alone to convert to a revived Arianism during the general conversion of Scandinavia to Catholicism seems insanely unreasonable to me. Who preaches it, with all the historic Arian realms gone under and left behind?
It would make some sense if they had simply adopted early and kept Arianism all along, but that puts them in pagan Scandinavia as Christians who don't even have the support of the Catholic Frankish empire to the south and this is a POD going back to somewhere between the mid-4th and late 7th centuries. And would totally transform the Viking Era into something unrecognizable.