If the objective lense is bigger it will take in more light...
This is correct, the front objective takes in the light, but watch ILya's video and you'll see how light is then directed through the tube.
a larger tube mean more light will pass through also.
This is incorrect, see the quote from the Swarovski forum I posted above, how much light that actually passes through the tube is not necessarily determined by the tube size, it is an optical principal based on the exit pupil and how wide our own eye's pupil can dilate. Can a 150mm objective gather more light than a 56mm objective, yes, but what's the point when the exit pupil is waaaay above what our eye can take in (as Ron mentions in the Swarovski post, it is "wasted light" in essence; however, it can have an effect on eyebox which allows for more forgiveness when it comes to eye position), and the same goes with the tube, even the smallest we have today (a 1" (25.4mm) outer tube diameter) still has glass larger than the required 7mm for light to pass through. Here is the last paragraph from the article above:
The reason a 30mm main tube scope isn't inherently brighter than a 1-inch scope is because both carry internal lenses much larger than 7mm, so there is no loss of light through either. The reason some 30mm scopes appear to project brighter views is probably because they were built with the absolute finest materials and effective light transmission (how much light the scope passes through) determines brightness, is a product of the number of air-to-glass surfaces in the scope (the fewer the better) and the anti-reflection coatings on those lenses. (The more the better.)
During the day a 56mm objective scope pulls in a lot more light than our eye needs; however, we don't buy 56mm scopes for solely daytime use, there are other benefits to the larger objective, especially when it comes to low light situations.
Its the same principle with cameras... bigger the lense and aperture, the brighter a picture will be... its simple...
A camera lens is designed to direct light to a sensor whereas a rifle scope is designed to direct light to your eye, most camera sensors are far bigger than what our eye requires hence the need for larger diameter objectives, but again it's not the barrel of the lens that determines how much light hits that sensor rather it is the front objective.
One final thought, the human eye is a very poor comparative detector to brightness change because of our brains ability to communicate what our eye actually picks up, this is why I always recommend to perform scope tests in poor lighting condition (after the sun goes down), this is when our eyes are at their worst and it allows us to identify flaws within a scopes design much more easily than during the day - I have seen many scopes perform very well during normal daytime conditions and fall apart when the light gets low.
PS - I am not an optical engineer so understand this information is what I've gleaned in my passion for long range shooting over the years but I am prone to error at times and always willing to be corrected.