You're asking two diff Qs here. You may or may not agree with the 3 liberal judges, but if you want to read their reasoning, I'm sure that's easy to find. Your title Q seems to be asking if they're right in their assessment, which may be veering towards the opinion-based. IIRC the SCOTUS majority judgement was that another judge has to determine whether anything the president does is in an official capacity or not.
Killing people with their own bare hands in a fit of rage, might not qualify as an official act. Ordering someone to be killed by the military may well qualify though [as official action], even if against Posse Comitatus, etc., e.g. on US soil. So the question, seems to me, is whether this broad presidential immunity for any official acts essentially leaves only impeachment as a recourse. And who'd be there to impeach him if he orders the military to arrest the opposition or the entire Congress? (And the military follows through. The only issue before a judge then would be whether such an order was official, not whether it was legal in any other sense, unless the act was first deemed unofficial. And it's hard to argue in any sensible way that POTUS ordering the military to do something is not an official act.)
FWTW, from press snippets:
Jackson warned that under the majority's "new Presidential accountability mode," a hypothetical president "who admits to having ordered the assassinations of his political rivals or critics...or one who indisputably instigates an unsuccessful coup...has a fair shot at getting immunity."
The chief justice dismissed the dissents, suggesting that his three liberal colleagues had misinterpreted the majority's opinion and were engaging in "fear mongering." Roberts argued that they "strike a tone of chilling doom that is wholly disproportionate to what the Court actually does today." He wrote that "like everyone else, the President is subject to prosecution in his unofficial capacity."
Furthermore, Chief justice Roberts put under the clout of official acts almost any kind of talk of POTUS with/to his underlings.
But Chief Justice John Roberts, writing for the majority, preemptively determined that Trump’s interactions with justice department officials were official acts because they are part of the executive branch and answer to the president.
Roberts also determined that Trump’s interactions with Pence were presumptively immune, since the president discussing responsibilities with the vice-president was an instance of official conduct. The burden was on prosecutors to prove otherwise, Roberts wrote.
So, POTUS telling Seal Team 6 to do this or that is very likely similarly construed as an official act, at least "presumptively".
Ibid (WaPo):
Sotomayor warned that a president is now immune from criminal prosecution if he orders the Navy’s SEAL Team 6 to assassinate a political rival, or if he organizes a coup to hold on to political power.