there are certain plants that have a finite lifespan built into them: they
go to seed and then they die. (Incidental question: does such a lifespan always
coincide with a single solar year?)

Not always. With some plants, like bromeliads, when they reach blooming size,
they bloom & set seed, then the parent plant dies, but first it produces
several baby plants around its edges.

What about trees? I can't recall ever hearing of a tree that died of old
age. They continue growing throughout their lives, but is there anything about
perpetual growth that necessarily leads to death?

With people, the obituary rarely reads that the person died of old age; he died
of cancer, pneumonia, heart-failure, or whatever other problem old people die
of.
The same with trees. They have various expected lifespans, from 15 years for a
poplar to 2000 for a bristlecone pine, but they rarely die of "old age." They
become weak & succumb to fungus or some other problem. What you are thinking of
has more to do with size. A large tree, like a redwood, can only pump water up
to a certain height. I don't know if this limit eventually leads to the plant's
demise.
Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)