But wouldn't the constant power relics also benefit from Inspiration/Zen? (I don't know if the damage proc relics are affected by inspiration.
Yes, but to a lesser degree.
Take a simplified example, suppose one relic gives a +4% passive bonus to base damage all the time, and another gives a +16% clicky bonus to base damage for 15 seconds every minute (1/4 of the time). On average (and in isolation), these both correspond to a 4% bonus.
Now combine these with an independent, Inspiration-like buff that increases all damage by 25% for 15 seconds every minute. (These aren't real bonuses for relics/buffs, but chosen so the results come out to be round numbers).
For the passive bonus relic, the overall increase will be +4% 3/4's of the time, and +5% 1/4 of the time (when the other buff is active). So the overall increase will be +4.25%.
For the clicky relic, if you stack the two buffs, then it is +20% 1/4 of the time, or +5% overall.