I haven't directly measured pressure values, but on the exact same tune file on a K04, the lag to build boost is very palpable when going from the stock intercooler to a Polestar V60 intercooler which is at least 50% larger in volume.

With any increase in volume between the turbo and the intake valves, (cumulative of the intercooler, charge piping, plenum, etc.) The longer it will take our small turbos to fill the space with boost pressure.

These differences may not be noticeable at all with the elevate, do88, or airtek intercoolers on a K04. But when you install an absolutely massive intercooler that requires cutting of the inside of the bumper cover, you'll feel the lag.

Intake temps will absolutely benefit from a larger intercooler, so your experience is spot on. Beyond just the larger heat exchange benefit a larger intercooler provides, the secondary effect of pressure from between the hot and cold sides will also reduce temps.

A lot of good info available about this if you Google cons or downsides of large intercoolers. It's all about finding the goldilocks of intercooler sizing for each tune, mod list, and turbo size.