Oh look, a buried lede:

They noted that the babies exposed to more screen time were children of younger first-time mothers who had lower household incomes, lower maternal education levels, and suffered from postpartum depression.

There are plausible mechanisms of action proposed here but their >2 hours/day sub-samples were small, baseline risk of developmental delay very low (5%), and it is essentially impossible to control for all the factors which are associated with more screen time. The effects they're reporting here (of the order of 5% absolute difference) are not large compared to the influence of socioeconomic factors on these outcomes.

Full paper here with free pdf download for nicer format.

So, for all the parents out there who have little choice but to have a screen keep your kids occupied while you deal with everything else life is throwing at you, don't let this report load you up with more guilt. Get some quality time with them when you can and choose children's programming which uses and teaches proper language (Sesame Street, not the Teletubbies). But you're doing little to no harm however you choose to manage it.