It is likely this is false for a general Jordan algebra, even finite dimensional over $\mathbb R.$ I downloaded the review of Faraut and Kuranyi by Kenneth Gross, and he emphasizes in a footnote he says:
What is now called a symmetric cone in older terminology was referred to as a domain of positivity, and what is now known as a Euclidean Jordan algebra was called a formally real Jordan algebra.
Now, from page 3 of McCrimmon's book, we find
A Jordan algebra is formally real if
$ x_1^2 + \cdots + x_n^2 = 0 \; \Longrightarrow \; x_1 = \cdots = x_n = 0. $
This is A Taste of Jordan Algebras by Kevin McCrimmon, about 2003.
So, my advice is to see if you can prove your cone property with formal reality as an axiom. As a matter of taste in terminology, I don't see how you can call something a domain of positivity if the sum of strictly positive elements can be zero. If it does not work out, you have several people to ask for a counterexample, that is showing what goes wrong when not formally real.
P S I think you will like the description of Euclidean Jordan algebras in Newton's Algorithm in Euclidean Jordan Algebras, with Applications to Robotics by Uwe Helmke, Sandra Ricardo, Shintaro Yoshizawa.