Why not nodes just keep track of of bitcoins and where they are? Like so:
Alice wants to send Bob 1 btc she does so and alerts a fair amount of nodes that she did this. The nodes then collectively agree that bitcoin#1251262613616 is now under the control of whatever address alice sent to.
What if Alice also sends the coin to Dave? Now both Dave and Bob are owners according to a "fair" amount of the network.
The chain is the record. Each block is basically a list of transactions. Each transaction moves coins from the control of some people to the control of other people (splitting and combining is also possible).
The coins are actually transaction outputs. Each transaction output has a value and a rule for spending it.