Yes very true , but it really depends on the engine too. Some go into relief just off idle. Others don't until 5000rpm etc. So different viscosities behave differently between engines. For example a 10W60 might go into relief at 4000rpm, but a 10W30 in the same motor might not go into relief until 6000rpm. So the latter oil would provide more flow for the same pressure.
Whilst the fuel economy benefits between thick and thin are small, they are definitely there but we're not talking say a trend you could see just over 1 tankful. For example, an ILSAC GF-4 rated oil must show 2% (IIRC) better fuel economy than a refererence fully synthetic 10W30. Moreover, many manufacturers in the USA are moving to 5W20 purely for fuel economy reasons. I believe people who would benefit the most from a thinner oil is one who does frequent short trips, where the cold viscosity differences between a 10W60 and a 10W30 are very large.
However I wouldn't say a thick oil would make you loose power compared to a thin on the dyno as the differences are down to the margin of error.
Cheers