i find that when this topic is discussed it tends to treat the melting glaciers as an infinite input.
if the concern around the shifts in the gulf stream are mostly long-term in scope, it seems reasonable to point out that greenland will eventually run out of ice, allowing the other processes that are increasing the temperature of the ocean to take control. even if the gulfstream collapses entirely, the general effects of sea temperature increases in the long run could lead to broadly warmer oceans in europe.
in theory, yes - this is something to be concerned about. where i am, the weather is often determined by a struggle between warm atlantic ocean temperatures and expanding polar air masses, the latter phenomenon being a mostly solar process. so, we also rely on warm atlantic temperatures for a lot of the heat in canada, especially during the parts of the year when we are tilted away from the sun.
but, from what i can see in front of me, what's actually happening is that the effects of global warming further south in the atlantic are dominating ocean temperatures off of the coast of florida, and it's the latter process that seems likely to be dominant, long to medium term.
https://www.theweathernetwork.com/ca/news/article/florida-current-is-weaker-than-it-has-been-in-over-100-years-oceans-climate-change