The on-use gives more damage sustained too, you are only losing damage when you don't activate it when it is ready or if you can't attack for a majority of the activation, which on the fights where that will happen there are parts which need more burst which make it worthwhile still. There isn't a single fight in the new content I wouldn't bring it to.
I suppose - I believe the on-use works out to 134.5 power if you managed to pop it exactly every 2 minutes (350/4+47). WH is 113 and the newer one is going to be 120. So it is a gain in power of approx 21.5-14.5. Rule of thumb says power is .23 DPS so that is 4.945-3.335 DPS. Again, this means you are successfully hitting it every 2 minutes exactly. So if you are at a utilization rate of approximately 20.86% you have a higher sustained DPS. A Utilization rate of 20.86% means that if you click the relic immediately to start the fight - you have 143.84 seconds to click it again before its DPS is lower.
Note: I hadn't run the math, so you are probably correct here.
I'll also note that I don't have an extra 300 daily comms right now - and given the choice of the click relic and the proc relic, as a balance sage, proc is better.
Again - both have 47 power so you have 87.5 power on perfect utilization vs 210 internal damage (bypasses armor) every 6 seconds. 87.5 X .23 = 20.125 DPS. 210/6 = 35. So proc is a clear 15 DPS over on-use.
If you can afford both, get both, though.