In the simplest terms, if all of your coins have been received to the same address and you continue to send your change back to that same address, owning the private key for that address is crucial if you want to spend them
I would never recommend that anyone do that, but yes, if you always use the same address for receiving bitcoin and you always send the change back to the same address then "owning the private key for that address is crucial". Also, making sure that nobody else has access to the private key for that address is crucial.
so essentially, the coins have pooled together.
I guess that depends on what you mean by "pooled together".
Lets use an analogy...
A standard "bar of gold" is 12.4 kg.
If I send you 10 nuggets of pure gold that each weigh 1.24kg and you toss them all in a bucket as you receive them, you have ten 1.24kg nuggets in a bucket. Are the "pooled togther"? If you want to spend them, on something that costs 6.2kg of gold, you'll still have to hand over five separate nuggets. (This is what happens when you
receive separate bitcoin transactions. Each "nugget" is an output from a separate transaction and the "bucket" is your bitcoin address associated with those unspent outputs). This is the sort of "pooling together" that happens when you
receive separate transactions to the same address.
Now instead, if you want to buy something that costs 1.86kg of gold, you'll can melt together 2 of the nuggets (for a total of 2.48kg) and pour them into a 1.86kg sized mold and a 0.62 sized mold. You can then toss the 0.62 sized chunk back into your bucket with the other eight 1.24kg sized pieces and give the 1.86kg sized piece in payment. You now have in your bucket 8 pieces that are 1.24 kg in size, and 1 piece that is 0.62kg in size. Meanwhile the recipient of the transaction has a new 1.86kg sized piece in his bucket. (This is what happens when you
send a bitcoin transaction. Individual "nuggets", previously received outputs, are selected from the buckets. They are listed separately as inputs to the transaction (the transaction is the "melter" that destroys the original inputs and lets you "pour" out newly formed outputs of new sizes). This is the sort of "pooling together" that happens when you
send a transaction.
Notice in the example, if you have 10 separate buckets, or just one bucket, it doesn't change the fact that you still have 10 separate lumps of gold. A good wallet can easily keep track of as many buckets as you have outputs.
If this were not the case, how would this transaction make use of an address which contains more than 50 coins (given that the initial mining reward was 50 coins)?
txid:c27212a482cc4add5cbc51f131c1bd8fd3dc7d1fb058724f2268d819ddef3f07
That address has never been re-used. It has (as of the time of writing this post) only ever received a single transaction output and has never received any change from itself.
The creator of that transaction previously received a single 151.55267355 BTC output from someone. They created a bucket (bitcoin address) that they named "1HTe9PK5wZSp2qXqn1yLFegGxj2Z3biW1X". Then when the sender melted together some of their own inputs to create a transaction, the sender poured out a single 151.55267355 BTC nugget that got tossed into that single "1HTe9PK5wZSp2qXqn1yLFegGxj2Z3biW1X" bucket. Now when the owner of the 1HTe9PK5wZSp2qXqn1yLFegGxj2Z3biW1X bucket wants to spend some of that 151.55267355 BTC output, they created their own transaction that melted down and destroyed the 151.55267355 BTC output, and poured out two new outputs. One was 0.3816135 BTC (which perhaps was sent to the recipient and stored in their own bucket labeled "1GKa5vmFXECfWQMF3C5rzufR2sto6e8r6a"), and the other was a 151.17096005 BTC output (perhaps of "change" that was tossed into a NEW bucket labeled "1PStiqfcYGkUu6p5QU3PCjmCdE1Ec744KC").
