It seems that the answer is yes. A MathSciNet search brought up the paper
Y. Berkovich, D. Chillag, and M. Herzog, Finite groups in which the degrees of the nonlinear irreducible characters are distinct, Proc. Amer. Math. Soc. 115 (1992), 955–959.
In it you can find a characterization of groups whose nonlinear irreducible characters have distinct degrees. In particular, such a group can't be perfect (see Lemma 1), and so will always have multiple linear characters as was noted in the OP. The proof, however, relies on the classification of finite simple groups, so it's is not "easy".
Addendum: I took a closer look at the related literature and happened across the following interesting result, which I figured was worth sharing. (It can also be used to give an affirmative answer to the original question.)
$G$be a nontrivial finite group. If the character table of
$G$has a column or row containing distinct rational entries, then
$G$must be isomorphic to either
The reference is
M. Bianchi, D. Chillag, A. Gillio, Finite groups with many values in a column or a row of the character table, Publ. Math. Debrecen 69 (2006), no. 3, 281–290.
The result from the classification of finite simple groups used in the Berkovich–Chillag–Herzig Berkovich–Chillag–Herzog paper is also used here (in very much the same spirit).