Like I said, unless it is a cheap one. $100 for a 50 mm scope is pretty darn cheap.
You are right to a point. Technically a larger objective has the potential to "gather" more light. However human eyes pretty much suck and can only use so much light to begin with. A larger objective will give you a larger exit pupil diameter, then again so will a lower magnification. However your eye can only use so much exit pupil before you are pretty much wasting it, 5 mm is about the norm in most cases from what I understand. So if your scope puts out 4-5 mm of exit pupil at its highest power you should be fine in any legal hunting light.
More important than objective size and exit pupil size are the quality of the lenses and scope. Being able to gather a lot of light means nothing if it can not transmit it accurately without a ton of light loss and reflection. When you look through a nice clear, crisp and what seems to be a bright scope your most likely seeing resolution, not brightness.
A cheap scope with a large objective will not transmit as much light or be as clear and crisp as a more expensive scope with a smaller objective. The better scope has much better glass, coated optics and a better internal design which GREATLY effects how it looks. There really is no substitute for quality glass. Getting a bigger objective will NOT make up for using cheap crappy optics. And you don't need to spend a butt load of money go get it, you just need to be frugal is all. You can get very good hunting optics in the $200 dollar range as long as you don't want a huge scope with lots of magnification and other bells and whistles that you probably don't need anyway.
Lets take a look at your typical bargain hunting scope. A $100 dollar 3-9x50 scope. It has a 5.5 exit pupil at 9X, a 15 ft field of view at 100 yards and weighs over a pound. Not to mention it will sit about 2 inches above your bore depending on what mounts you use. I also highly doubt a scope of that size in that price range is going to have decent optics or any coatings on any of the lenses. There is also the issue of dependability and how long it will hold it's zero. Especially considering a heavier scope with large lenses is more susceptible to recoil because of its increased mass. And when it goes bad you might as well just pitch it. And hope it doesn't do it on the hunt of a life time or when that monster buck you have been chasing all year is in front of you.
Now compare that with a more expensive more compact scope. Lets say a Leupold VX-1 in 2-7x33. It has a near 5mm exit pupil at 7x, 17.3 ft FOV of 100 yards and weighs 10 ounces. It also mounts much lower to the bore of the rifle for more consistent shooting and has more eye relief.
It also comes with excellent optics, top notch quality control and fully coated lenses with very good light transmission. Even though this scope is smaller it will be MUCH brighter and more crisp in every condition compared to a 100 dollar 50mm bargain scope. The difference would be like looking through a piece of plexi glass vrs expensive precision ground prescription glasses with every available coating. And this scope only costs $200.
Not to mention the incredible warranty they come with. If you beat it against a tree and were the 3rd owner they would probably still replace it for you.
I have a $400 dollar varmint scope that is 20X with a 44 mm objective. I would be willing to bet it looks twice as good at 20 power as your cheap 50 mm scope does set at 3 power. Quality glass really does make that big of a difference. And once you step up to the Zeiss Conquest or the $500 and up scopes they look even better yet, especially in low light conditions and higher powers.
Next time you are in a gun shop ask to see a cheap scope, then compare it with a more expensive one of similar or smaller size. Make sure they are adjusted properly though, and if possible look at something farther away. If you can't see a drastic difference you might want to consider getting your eyes checked
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Normally I end my posts by saying this is just my opinion. However this time it is not just my opinion, it is pretty much factually true.
I'm not saying a cheap scope won't work, they do, I have used them. I'm just saying a better scope really is better, especially in bad weather and harsh conditions.
Paul