LsPush is superior in many ways to LsClient. What LsClient basically does is tell your Lansweeper server that the computer is online and wants to be scanned, while LsPush performs all scanning actions and commands locally. This allows it to use far less bandwidth and gives the ability to scan computers that do not have a direct connection to your Lansweeper server, which LsClient cannot do. LsClient is mostly a legacy executable at this point, in most cases its roles have been taken over by LsPush. Only in the case of "After Scan" would LsClient indeed be more useful.
As explained by my colleague, LsPush doesn't work with "After Scan" deployments by design, as LsPush is in most cases used due to connection issues between your scanning server and the computer in question. A scan completed by the scanning server itself ensures a connection capable of deployments. The level of certainty is key here, as we strive to provide an as error-free experience as possible.
We've noted your request though, and have added "LsPush After Scan deployment" to our customer wishlist. At this moment though we unfortunately cannot guarantee that this feature will be added and cannot provide you with an estimated release date.