First of all, let's start by stating that there's not just a single type of immortality. Being immortal often doesn't mean being unkillable. It just means you won't die due to natural causes, be it illness or the passage of time. This is the first and more common type of immortality.
The second type of immortality is one where you are, really virtually unkillable, be it due to an insane fast regeneration speed or extreme durability. This would mean that not only would you have an infinite lifespan, you'd also survive pretty much everything except the heat death of the universe(which would never happen in this case since as long as you live, the universe will never be able to reach maximum entropy, therefore never dying either).
Now the common arguments for immortality being a curse are the following:
->You'd outlive everyone you care about, and would be lonely.
->You'd end up floating through the universe whenever the earth died and would go mad.
People make these arguments because they are judging immortality through a mortal lenses. As for the case of outliving everyone, the same can be said for mortals. Parents outlive their children all the time, orphans outlive their entire family all the time. Do they despair? Sure, temporarily. But they eventually move on and meet new people. They don't forget their family but they also don't dwell on the past. If mortals are capable of this, an immortal would too. And since they have infinite time, they could despair for thousands of years and still move on afterwards. And these thousands of years would probably go by really fast for the way an immortal perceives time is completely different from ours. A thousand years is merely a speck of dust in the eternal sands of time. Which brings us to reason number 2. In this case, sure you'd float in space, but only temporarily. Eventually everything comes back around, everything changes, except you. Earth is destroyed but another planet is born. It might take millions of years but that's nothing to you. You have eternity ahead of you. Sure, you'd go mad. But after a million years you'd get bored and go sane again. You are eternal, you'll live as long as the universe. There's no reason to despair for you know there's time for everything. And for those millions of years you're waiting for a new world to form, you get lost in your memories. You remember your family, your friends, their kids, grandkids and descendants. You sleep through planets collapsing and stars exploding. And you are peaceful, for as long as you're immortal, anger and sadness always subside with time. You're happy as you look forward to the next planet, the next race you'll meet. The wonders they'll show you. And, the wonders you'll show them.