A 500W PSU won't necessarily draw more than a 250W PSU, that is merely its maximum sustained load (what the rest of the system asks for) rating. The Bronze 80+ rating is likely part of the problem here, that indicates what the power draw from the wall is compared to what is being provided to your system. Titanium 80+ would net you about 10% reduction in wall power usage. Keep in mind that manufacturers play fast and loose with the certification process and a consumer unit may not actually be what it says on the box, you need to rely on quantitative reviews.
Other than that, spend some time in the firmware settings. Powtop also does a great job at shaving off some watts.
"A 500W PSU won't necessarily draw more than a 250W PSU"
Mostly true, but not exactly. Most computer PSUs are more efficient when operating around 50% of their rated load. So if a computer consumes 125W internally, a 250W PSU would translate to lower power consumption measured at the wall than a 500W PSU, typically by about 2-5%.
For example see the chart https://www.sunpower-uk.com/files/2014/07/What-is-Effciency.... (115 VAC input) : 88% efficiency at 25% load, vs 90.5% efficiency at 50% load. In practice if the consumption is 125W at the PSU's DC output, this translates respectively to 142W vs 138W measured at the wall.
This 2-5% difference may not seem much, but it's similar to upgrading 1 or 2 levels in the 80 PLUS ratings (Bronze, to Silver, to Gold, to Platinum, to Titanium).
Switch-mode PSUs are very inefficient at the low end of their duty cycle.
A 250W 80-bronze PSU for a 60W load will be operating at 25% capacity and 82% efficiency or better.
A 500W 80-titanium PSU at 60W will be at around 12% and 90% efficiency or better.
So, an 8% difference in minimum required efficiency...for a huge increase in cost.
It's much better to buy a high "tier" PSU (for reliability and safety), sized so that it spends most of its time at or above 20% duty cycle (which in OP's case would indeed be 250W.)
80-gold is very common in the marketplace and where most people should probably be buying.
For building NAS, there's a myth that you need good enough PSU to spin up all disks at once (unless you have delay mechanism). I wonder is it still matter now.
A 500W PSU won't necessarily draw more than a 250W PSU, that is merely its maximum sustained load (what the rest of the system asks for) rating. The Bronze 80+ rating is likely part of the problem here, that indicates what the power draw from the wall is compared to what is being provided to your system. Titanium 80+ would net you about 10% reduction in wall power usage. Keep in mind that manufacturers play fast and loose with the certification process and a consumer unit may not actually be what it says on the box, you need to rely on quantitative reviews.
Other than that, spend some time in the firmware settings. Powtop also does a great job at shaving off some watts.