I think achievements are fun. I don't care about gamerscore or bragging rights or anything like that -- the thing about achievements is that functionally no one cares about your achievements other than yourself. But I don't think that means they're worthless, and the idea that they are just because they don't offer tangible rewards is really reductive. "Playing video games" in general doesn't offer anything except "experiences" and achievements are just another method of delivering experiences. I don't think a largely ignorable feature should go away just because some players don't like them.
Now, limited continues? It really depends. They're just another part of the game design language. I don't think they work for most games, but some game design is functionally incompatible with unlimited continues. Most instant-respawn shoot 'em ups just don't work without some mechanism to halt progress. I don't know how many amateur reviews I've seen over the years which have completely written off tightly designed games like Ikaruga because it can be beaten in 30 minutes if you credit feed. Would it send a different message if you only had a few continues to work with? Or no continues? You could argue that the issue here is entirely within the realm of player expectation, but I'd argue that it's up to the game design to guide those expectations.
This also extends to games like ZeroRanger, where the game's artistic themes are tied to the limited continues, as is a particularly pivotal moment at the end of the game. Is it right to say "x mechanic isn't fun, so there's no reason to use x mechanic" if that mechanic is being used to server a purpose other than "fun"?
Now... how to people feel about lives systems with unlimited continues? Like the Mega Man games, for example? I really like the gameplay cycle in the classic games -- you have a few attempts to make it through the stage and clear out the boss. If you die a lot on the stage, you have fewer attempts at the boss, potentially with limited resources. If you fail, you have to make another run at the stage or look for a boss that you can beat. If you can beat a boss, you might be able to use the resources you obtained to make it through the boss you previously failed against.
On the other hand... they completely screwed this up in the later PS Mega Man X games. You have lives, but you have unlimited continues. Continuing just takes you back to the same checkpoint that dying sends you to. And you can't escape the stage without having already cleared it; the only way to leave is to lose all your lives and choose to return to the stage select. And on top of this, they hand out lives like candy. Every reploid you rescue in the stages gives you an extra life, and the stages are infested with helpless reploids. In effect, lives in Mega Man X5 are nothing more than a barrier keeping you from going back to the stage select.
It's a complete failure of game design, and is perfectly emblematic of the design quality of those games.