An often used intuitive reason why the limit is 1
is that, as stated above,
from the unit circle definition,
the length of the arc of a small angle
is about equal to the height if the triangle.
However, if you do a similar thing
with a right angled isosceles triangle
with short side 1
by putting smaller and smaller
similar triangles on the hypotenuse,
the sum of the lengths of the legs of the small triangles
stays 2, no matter how small the triangles are
and how close their right angle vertices
approach the hypotenuse.
(Wow - that is a mouthful.)
The main conclusions, imho,
are that intuition can be led astray
and that, to rigorously prove the
limit exists and is 1
takes a more precise definition of sin.
Math often seems to work that way.
A picture is drawn, or some
plausible algebraic manipulation is done
(like considering a power series beyond its radius of convergence)
and something reasonable is deduced.
However, to prove it in a
satisfactorily rigorously way
often requires replacing the original,
intuitive definition with a more precise definition.
Anyway, that's how it looks to me.