One thing that I have used is an inrush current limiter. It works, but there are few options on the market that are pre made and relatively affordable. An auto bypass relay that selects between UPS and outlet power also does wonders for maintaining power to your computer(s).
The size of your power supply matters. My server only draws 200w under load yet has a 1400w power supply. My 700va, 450w UPS would trip on start up as the inrush for the server was too large. I upgraded to a 1500va, 1150w UPS and it's been fine ever since. Both UPSs were Eaton 5SLCD models.
I am sure it does, but it is a bit weird that the old APC handled it fine (and to be honest, should the ups trip when the inrush current is to large when connected to power??
First part:. Different UPSs have different behavior. Some are more sensitive to disturbances, whether caused by line or load.
Second part, the UPS being connected to utility has no effect on the inrush to the server. It all depends on where in the AC sine wave the energy starts flowing. Ever plug something into an extension cord with a clear receptacle? Sometimes there is a sizable blue arc, sometimes small, sometimes none. All about where in the cycle the connection is made.
What Eaton unit are you using?
I run hand full of Eaton 5P 750va in satellite network racks in the building. Those are good to 600w power delivery. No issues with them.
Put a meter like a Kill-a-Watt on that system and watch the power draw on start up. Might be worth hooking up another system to it as well and see if the problem persists. If it doesn't act up on the 2nd system, might be an issue with the PSUs for the 1st system and what killed the APC.
You seem to have a sample size of one with Eaton. What did Eaton Technical Support have to say? The last time I called them they were very easy to work with and very helpful. If you bought a used Eaton, you probably got a bad unit.
It is new, but you might be right, I have just been unlucky.