Your typical UPS (i.e., excluding a double-conversion UPS), including all APC SmartUPS that I've worked with in the past, does nothing to the power except when running on batteries. It will draw a little extra power in regular use. When power is restored after an outage, it will draw extra power to charge the batteries. During a power outage they will just monitor the incoming power and wait until it is stable and back in range (e.g., 105V - 125V) before switching from battery to line power. If there is a room/building UPS, your own UPS will never switch to battery power. (Two exceptions: if everything is totally out (in which case, in my opinion, it doesn't matter much because you have no internet connection) or if you have a separate non-UPS power source - i.e., the data center provides both backed up and non-backed up power to your rack. That last possibility seems a bit odd to me, but based on comments it could be a thing in some places.)
So the end result is that from a practical standpoint - energy usage, interference, etc. your UPS is effectively dead-weight if there is a room/building UPS (which itself is hopefully backup by a generator for long-term outages). The only real advantage to your own UPS is that you can setup monitoring (APC has this as a standard feature) to do an orderly shutdown, but that will only apply if the room/building UPS stops working.
However, safety is an important thing. The big red shutdown button (or equivalent) for use in a true emergency - e.g., fire or flood - will shutdown power to everything else but not to your computers. Which means if your rack is part of a fire then you have a serious problem, because power will keep going. Or if everything is getting flooded, you are now pumping power into the rising waters when everything else has been safely shut off.
Plus, unlike your own office where the rack might contain your router, fiber/cable modem, etc., in a data center you are dependent on the provided network connections. If power goes out to everything else, continuing to run your computer might save some data corruption but won't actually keep your web server accessible.
So follow their directions - even though their stated reasons are meaningless.