The motherboard does affect performance but not in the same way a different CPU/GPU does. Most gains come from overclockability/feature set rather than straight speed.
Yeah, that's not accurate either. Not really. A poorer quality motherboard, which will of course be cheaper (Hence, directly related to COST), will or can absolutely have an effect on stability and boost behavior, not to mention PCI bus performance (Cheaper boards just DON'T have the same quality or performance, over ANY of the buses, because they are, simply put, not built as well and don't incorporate the same high quality components such as quality chokes, caps, switches, diodes, etc.) and also in almost ALL cases, the thickness of the PCB itself. Thin PCB tends to lead to an increased probability of early failure from flexion, which can and will allow a higher rate of damage to traces, solder points and other forms of breakage on the motherboard.
Since other components MUST rely on the motherboard, in order for them to have any value at all, it therefore stands to reason that in order for those components to perform at the highest possible level of performance that they are ABLE to, they must have clean, stable power delivery from the power supply in ample quantity, and a motherboard that is able to handle that power delivery and the demands of the hardware either by weight or electrical stress without breaking, throttling or otherwise dampening the performance of the hardware.
There is a REASON why many motherboard reviews include comparisons to other motherboards while performing the same tasks with otherwise identical hardware, and that reason IS because there are differences from one board to the next in how well they perform those tasks. They would not waste their time doing those comparisons if there were no fruit to be had for doing so.