Is the generic one suitable for the latest laptops? I've seen some laptops don't charge unless you get the correct charger
Yes. As long as the rating on the generic charger matches the one of your current charger, and the charging pin is the same size with the same polarity (negative inside & positive outside, or vice versa), then you're good to go.
Regarding the rating, the output voltage should be exactly the same and the output current (amps) should be at least the same as your current charger, or more.
There's nothing more to it than that.