Gotta love threads like this. I was about 10 or 11 when this band hit the scene, maybe 12. I remember them, and I remember not paying attention to them because they weren't the kind of heavy rock I was into, but I will never forget that woman's voice, that kind of airhead act with that squeaking noise she makes between lines is so iconic.
I am not overly nostalgic, but watching these videos definitely sends a wave of 'member berries over me. It prompts me to explore lots of 80s music I missed on the first pass.
As for it sounding "dated", I play in an originals band with younger dudes and we and the other originals bands I've seen these days are definitely into referencing 80s post-punk and new wave sounds. If you search around youtube or bandcamp a bit you will find tons of bands with tags like "coldwave" and "post punk" that are basically guys and girls in their 20s making essentially 80s music, and some of it sounds awesome. Boy Harsher, Soft Moon, Naut, too many to list.
The post-MTV Internet world we live in now is a whole different place; music you thought was forgotten about decades ago is being kept alive and well and spread out geographically. There are doom metal bands referencing the 70s, psychedelic bands referencing the 60s, Southern rock bands, crazy jazz bands, you name it, it's out there and it's alive somewhere. The decline of the "rock star" means that nothing and everything is cool at the same time. Confusing and strange and hard to make money at perhaps, but great for creativity.
Anyhoo, love Missing Persons! Another band I wish I paid more attention to back in the day when I was obsessing over Rush and Iron Maiden!