Why check for multiple nonces despite finding a golden nonce? I'm still new at trying to understand the entire idea surrounding the mining process.
I read the following:

When mining, you are given a set of data to perform hashes on. As you
noted, this is done by manipulating the nonce and hashing each time
the nonce changes. What you really want to do is check all possible
nonces. Even if you've already found a &quot;golden nonce&quot; (one which gives
you a hash starting with 32 zeros), you need to keep searching for
more. There could be anywhere between 0 and 2^32 solutions to a given
block of work, so it is in your best interest to keep looking for
more. Hence, there are no stop conditions in the sense of when to stop
running your algorithm, other than having exhausted all possible
nonces (at which point, you would get more work).

Source
This confused me, because I thought that once you found a golden nonce, you can broadcast that to the network and move on. Why would you continue to check for more? Is it because if you let's say found 5 golden nonces and broadcasted all of them as you found them, you'd have a higher probability that the next block would be building off the one you found? Adding onto that, if many people find valid nonces to a block, how is the winner of the reward, and which gets included in the main chain, determined?
There's a difference between what is called a "golden nonce" in the context of the source and a valid block header hash.

To reduce payout variance, miners group into pools and share their income proportionally to their hashpower. 

To determine each miner's hashpower, pools use a share system: when you are mining and find a block header of high difficulty (with lots of heading zeroes) but not high enough to be above the current target, you send it to the pool. These difficult-but-not-enough headers are called "shares".

Since finding them is way more frequent than blocks (miners often send several of them per second), no matter how small your hashpower is, it will still be taken in account at payout time even if you're unlucky. The more shares you send, the more hashpower you have, the bigger the share of the pool's income you will get.

They are not broadcast to the bitcoin network, since they would be immediately rejected there and are of no value to anyone but the miner that found them and the pool software.