Often there are tradeoffs. E.g. when I put my phone in powersaving mode the dimmed screen is very hard to read in bright sunlight. My browser not processing background tabs might be a useful feature sometimes, but when I'm listening to music on Youtube it would be a real inconvenience.
Desktop Safari doesn't have a 'power saving mode' and it consumer less power than Chrome. Apparently MS Edge is the same (I don't use Windows so I can't know). These browsers don't have user-visible tradeoffs, they're just more power efficient.
Or is this just a case of Chrome being really really unoptimised and power hungry?
I have no idea about specific browser implementations. I know I perceive Safari and Edge as less responsive than Chrome, which could easily be due to tradeoffs they make for power consumption.