Have a look @ Faliraki. No longer full of larger louts.
Pefkos or Lardos would be ok.
Not much package accomodation now in Lindos Village. More up on the Krana area, but involves walking up & down cardiac hill to the village & beach.
Ixia & other west coast resorts are windy.

Depending how old the kids are, i would definitely recommend Pefkos because it is so easy to get around. Regarding apartments, it is hilly in parts so you would need to look on line at various places that are not up a hill.

Have a look @ Faliraki. No longer full of larger louts.

It never really was,aside from a minority in the bars at the top/middle of bar street in peak season,you get that in jus about any big resort,would even consider comparing Faliraki with somewhere like Magaluf or Benidorm when it comes to rowdiness,i doubt you would yet thousands of families go to these places without incident and have done so for decades.

To the OP,as said,depends how old your kids are,what attitude you have towards entertainment etc,if you want a huge beach with every amenity imaginable,an ultra-modern waterpark on the doorstep and bars,restaurants, hotels,apartments and studios that mostly cater for couples and families(despite the stereotypical media BS)then Faliraki is excellent. It's been 80% families since i can remember,the things you heard a few years refer purely to one very small area down one street in the resort,and even that part is full of families out strolling,shopping etc,same as it ever was.

That said Pefkos is also very nice but there's not as much to do for families with older kids