People don't tend to invest so much in stocks, instead settling for other instruments, like ISAs that already handle the tax end of things (https://en.wikipedia.org/wiki/Individual_Savings_Account). ISAs are an attempt by the UK government to encourage saving / investment by providing tax incentives to do so. Again, tax considerations done upfront vs after-the-fact.
If you do invest in stocks, you're liable for capital gains tax, just like in the US, but whether you need to file or not is dependent on how much you actually made. If you make too much, you'll have to file taxes yourself, but it's relatively straightforward.
The point is not that the system is perfect, but that it handles the case for probably >90% of the population, without stress or complications for them.
There is no personal benefit.
For cash gifts the charity can claim back 25p in the £ on your gift. For property, land and shares you deduct it against your tax payments at the end of the tax year and claim for a refund.
Every individual gets assigned a tax code, and that tax code defines how much is taken from your pay. That tax code takes in to account how many kids you have etc. etc.
The UK tends to approach tax deductions in a different way, preferring to subsidise stuff up front instead, so you never have to remember to file or figure out what your deductions are. You never paid out the money in the first place.