Does Bitcoin have a built-in life expectancy?
but never actually get there 


Except at some point there will be a rounding error (as numbers are integers in the code) to zero new coins being mined.


  does this imply that the incentives for miners will also be reduced? 


Not really. First of all the miners also collect all the fees from all transactions. That should compensate for the reduction in new coins being mined. Another factor is the price of Bitcoin, either in USD or by the time USD doesn't exist anymore, the number of aircraft carriers you can buy for 1 Bitcoin.

Rolling out a new chain will actually probably become harder and harder over time as the network effect causes people to use the chain that is most useful. The number of other people that use it is a big part of its usefulness.