Why does not MOSEK allows a quadratic term in the objective for conic models?

If you have quadratic term in the objective, i.e. something like

$\begin{array}{lc} \displaystyle{} \mbox{minimize}&\displaystyle{} 0.5x^{T}Qx + c'x \\ \end{array}$

the problem is convex if and only if there is a matrix

$L$

such that

$Q = LL^{T}.$

This implies that the above problem can be rewritten as

$\begin{array}{lccl} \displaystyle{} \mbox{minimize}&\displaystyle{} t + c^{T} x &\displaystyle{} &\displaystyle{}\\ \displaystyle{} \mbox{subject to}&\displaystyle{} L x - y &\displaystyle{} = &\displaystyle{} 0, \\ \displaystyle{} &\displaystyle{} ||y|| \leq{} t &\displaystyle{} &\displaystyle{}\\ \end{array}$

This shows that any convex quadratic problem can be rewritten as a conic quadratic optimization problem. See also the MOSEK manuals for this and other reformulation tricks.

Hence, there is no loss of generality by requiring that a conic problem has linear objective, and there are in fact good reasons to require it.

A purely conic model is guaranteed to be convex, while a quadratic model may not be. Hence, for quadratic problems the convexity status should be explicitly checked.
A quadratic term in the objective destroys the simple and powerful conic duality theory.
The primal-dual algorithm for conic quadratic problems is the most powerful algorithm known for convex problems. Therefore, it is a wrong assumption that quadratic term is easier to handle than quadratic cone.