Well, first a note on terminology; that cast claims Riksrådet (usually translated as 'Council of the Realm') is the 'assembly', which was the Riks_dag_. It was in that era more akin to the Privy Council in Britain. Perhaps the confusion is because Riksrådet in that era would refer to their meetings as a 'senate'. But the meeting where Alexander Leslie's letter was read, there were (per the minutes) only four people in senatu, so hardly an 'assembly' by any standard.

I do not know the caster is coming from with the claim that Riksrådet 'approved' the letter from Alexander Leslie. What the minutes say for this meeting, that of June 28, 1638 (in my own translation from the heady mixture of Swedish and Latin of the era (termed 'Chancellery Swedish'):

1. The letter of Field-marshal Leslie was read, on the actions of the Scots, which appear to be perturbing the religion and freedom of the country.

This is the full extent of the discussion as recorded. If it provoked sympathies, it is not recorded in that record at least. It certainly did not provoke any official decision to side with the Covenanters, or even decision on whether to decide on that matter. Leslie is not mentioned in the council again until a month later (July 28) when he has formally requested to return to Scotland, and the matter is referred to the Chancellor Axel Oxenstierna for his opinion.

We do not know the exact contents of his letter(s). The fact that a copy of the convenant was included is only from a single note by Oxenstierna mentioning it (Exemplar illius confessionis et foederis in Rikskansleren Axel Oxenstiernas Skrifter och Brefvexling, second series, VII, 480-481).

What's clear from the minutes of later discussions, is that Leslie had demanded to be relieved of his service to the Swedish crown, in order to return to Scotland and fight. Alternately, Leslie asked for a temporary leave. He also demanded artillery, muskets, rifles and ammunition.

Oxenstierna was positive to supporting Leslie, as he recommended granting most of his requests, it seems. But, it is clear from both Oxenstierna's recommendation and the ensuing discussions, these weapons were to be seen as a private purchase and granted as a form of advance on his salary. The Swedish government was very concerned with avoiding the appearance of supporting insurrection against the Stuarts, even if their relationship was strained (but not as bad as it'd become in ensuing years). For the same reason the suggestion of a leave-of-absence was dismissed, as they did not want to have a rebel in their formal employ. There was also much discussion about whether they needed Leslie in the fighting down in Westphalia; Ultimately they did (as the podcast says) let him leave with 2,000 muskets and some cannon, but presumably less than Leslie had requested.

Now, getting to your question more specifically, one thing that does not show up at all in these discussions in the minutes of the Riksråd in 1638, is the actual cause of the Covenanters. There does not appear to be any particular religious sympathy with either side. Nor have historians who've studied the topic (e.g. Grosjean), to my knowledge, pointed to that as a reason for Swedish support.

This does not seem that strange. Sweden itself was, at the time, a staunchly Lutheran society. "Nonconformists" were simply not permitted to exist. It did not matter whether a Swedish citizen converted to Catholicism or Presbyterianism or Calvinism or anything else. The punishment for those who refused to recant was the same: Death. (And the common Lutheran confession did not stop Denmark from being considered Sweden's greatest enemy in the era)

Sweden's informal alliance with Scotland was rather based on personal contacts, on the loyalty and service that Scottish mercenaries had provided for Sweden for 60 years by that point (Leslie himself for 30), well back into the 16th century. Scots had made careers, some had become naturalized and even ennobled (hence Swedish noble families named Hamilton, Douglas, von Wright and Forbes among others). 15 were naturalized and ennobled just in 1648-1651. Johan Skytte, member of the Council (as was his son later), had a Scottish wife and his son Bengt who succeeded him on the council was thus half-Scottish. Queen Christina had a Scottish lady-in-waiting, Jane Ruthven, and so on and so forth.

Axel Oxenstierna himself, who was mostly careful and diplomatic, would eventually make his stance clear; he felt Charles was being tyrannical and unreasonable and should have sought a compromise that limited the power of the clergy. It is an opinion based more on the politics of religion and not the religion itself.

Another illuminating exchange is in the council records from a decade later, after the execution of Charles I, the exiled Charles II asks Sweden for ships and weapons on behalf of Scotland. Queen Christina, in attendance at the meeting February 29, 1649, asks if Sweden should not support Scotland, as it has done in the past. (somewhat overstating the case, as no formal support or alliance ever existed) Gabriel Oxenstierna (Axel's brother, advocating neutrality), retorted "One has merely supported a particular person, and nothing further.", exaggerating a bit in the opposite direction by casting the relation in purely personal terms. Field-Marshal Carl Gustaf Wrangel states that these issues all stem from Charles II's mother Henrietta Maria (implicitly; her Catholicism).

Christina brushes this off; stating outright: "Religion is only a pretext. For us [Sweden], it is all the same whether they [Scotland/England] are ruled by Papists or Calvinists."

Christina continue to argue in favor of supporting Scotland, but not on religious grounds. Sweden would ultimately not resolve to take any side in that conflict, and by the Battle of Dunbar the next year, were perhaps glad they didn't.

Ultimately the situation was such a diplomatic quagmire; navigating the waters between England, Scotland, Catholics, Covenanters, Puritans, Stuarts.. For all the pro-Scottish sentiment that existed, it was very difficult to even determine which side was 'pro-Scotland', with multiple parties sending envoys asking for support.

Edit: That said, Sweden did remain loyal to Alexander Leslie personally; Queen Christina (shortly beore abdicating) securing his freedom in 1654, even as Sweden was seeking an alliance with Cromwell at the time.

In short the actions were not pure political calculation, but the goodwill that existed towards Scotland was almost entirely rooted in the their influence and long (and recent) history of service to the Swedish crown at that point. Nothing seems to suggest sympathy for Presbyterianism, which is indeed more theologically distinct. (today the Anglican church and Church of Sweden are in fact in communion)

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Alexia Grosjean, An unofficial alliance: Scotland and Sweden 1569-1654, Brill, 2003

Svenska riksrådets protokoll, 7, 1637-1639, Nordstedts, 1912

Svenska riksrådets protokoll. 13, 1649.