Buy from the App Store via Desktop browser and have app auto-install to my iOS device?
It is not possible to install apps on your iOS devices by invoking the download using a Desktop browser.

Till recently (iTunes 12.6.3) it was possible to purchase an app on iTunes running on a Mac/Windows PC and have it automatically downloaded on linked iOS devices (The iOS device should have the setting enabled to auto-download newly purchased apps). With iTunes 12.7, the iOS App Store has been removed from iTunes for Desktop.

Thus, currently there is no way to push an app on an iOS device.

However, you can have a newly purchased (free/paid) app automatically downloaded on all your linked iOS devices, by going to Settings app → iTunes &amp; App Stores and toggling for Apps under Automatic Downloads.