There is a reason for studying colour theory, and unfortunately sometimes you can make things too simple.
There's nothing wrong in principle with having a bright, highly saturated colour for your brand, but using it as a background for a large area of a web page is risky. It's hard to achieve good contrast for any text you put on top, whether you go light or dark, and on some screens the background alone may become overpowering. It might have been better either to go with a significantly darker shade of blue or to desaturate the main colour for the large background area here.
Using a similarly bright and highly saturated but near-complementary colour as an accent right next to that main colour is another risky move. That sort of combination tends to clash, which is why you're probably getting eye strain looking at the CTA button on the demo page here. I suspect it would be difficult to find an ideal flat button colour based on that red accent once you've chosen the blue background, even if you played with the saturation and brightness. However, putting some sort of separator around the button, such as a border stroke in one of the neutral colours so you don't have the two clashing colours actually touching, would probably limit the damage.
The advice on using slightly colourised neutrals instead of plain greys is good, though.
The risk with highly saturated colors is they're more difficult to reproduce accurately in cross-media applications, namely print media.
Grays are easy to reproduce in cross-media applications, in theory. In practice, there's so much variation in displays and print, and human vision is sensitive to achromatic colors, the error is essentially guaranteed and it will be noticeable. So you're almost always better off with colors that aren't achromatic, aren't highly saturated, and aren't memory colors, like skin tones.
There's nothing wrong in principle with having a bright, highly saturated colour for your brand, but using it as a background for a large area of a web page is risky. It's hard to achieve good contrast for any text you put on top, whether you go light or dark, and on some screens the background alone may become overpowering. It might have been better either to go with a significantly darker shade of blue or to desaturate the main colour for the large background area here.
Using a similarly bright and highly saturated but near-complementary colour as an accent right next to that main colour is another risky move. That sort of combination tends to clash, which is why you're probably getting eye strain looking at the CTA button on the demo page here. I suspect it would be difficult to find an ideal flat button colour based on that red accent once you've chosen the blue background, even if you played with the saturation and brightness. However, putting some sort of separator around the button, such as a border stroke in one of the neutral colours so you don't have the two clashing colours actually touching, would probably limit the damage.
The advice on using slightly colourised neutrals instead of plain greys is good, though.