[kriggy]

- Could neopentane triol or trihalide react with ammonia to provide the triamine?

I would say yes but I suppose you get crazy mixture of products since after the amonia is alkylated first time, it is then more reactive towards another alkylation.
What about using gabriel reaction instead of direct alkylation?

[Enthalpy]

Thank you! Even with the kind of explanation that fits me, nice of you.
What about a large excess of NH₃? Since all OH or X must be substituted, there is no hard limit to the proportions. For ethyleneamines, producers do get ethylenediamine as well, in useable proportion influenced by the reactants.The wild mixture of products must of course be separated.

Variant of the azide way: the available full text of
"Ligand Synthesis and Metal Complex Formation of 1,2,3-Triaminopropane"
states on its second page:
"Due to the potential hazards of polyazides, a procedure was developed that did not require the isolation of these intermediates as pure substances. The solutions of the azides were rather transferred directly into the autoclave and hydrogenated to give the polyamines".
Any applicable to TAME instead of TRAP?

[kriggy]

That could work, It needs to be tried but might be easier than gabriel reaction. (Maybe even using liquid NH₃ as a solvent?  )

[OJM]

While the method of Fleischer has its safety drawbacks, the reagents used are cheap.

[Enthalpy]

1,1,1-Tris(chloromethyl)ethane is available at Sigma-Aldrich (81€/mL) and Alfa Aesar (97€/25g). Would that qualify as "cheap" for the synthesis of a ligand? For a rocket fuel definitely not.

The tribromo is far less common, no price given online.

How do they get the halo atoms at different carbons? Free radical halogenation would preferentially group them, or?