Unipotence is equivalent to unipotence of all finite-dimensional representations

Suppose $G$ is unipotent. If $V\mathrm{\ne }0$, take a composition series $0=V_0\subset \cdots \subset V_m=V$ as in [F2]. Each simple quotient $V_i\mathrm{/}{V}_{i-1}$ is a simple representation of the unipotent group $G$, hence one-dimensional with trivial action by [F1]; reading the flags in a basis adapted to it shows that $V_i$ is obtained from $V_{i-1}$ by adjoining a trivial line, and induction on $i$ puts $V$ in the unipotent form of [F1]. The first step $V_1$ is a nonzero fixed vector in $V$, so every nonzero $V$ has one.

Conversely suppose every finite-dimensional representation of $G$ is unipotent. Then every simple finite-dimensional representation $S$ is unipotent, and a unipotent simple representation has a nonzero fixed vector (the first step of its flag), so $S$ is one-dimensional with trivial $G$-action; since every rational representation is a union of finite-dimensional subrepresentations by [F2], every nonzero rational representation contains such a simple subrepresentation, hence a nonzero fixed vector. By [F1], $G$ is unipotent.

For the closure properties, let $V$ be a unipotent finite-dimensional representation with a flag $V=V_m\supseteq \cdots \supseteq V_0=0$ with trivial successive quotients. If $W\subseteq V$ is a $G$-stable subspace, the subspaces $W\cap V_i$ form a flag on $W$ with successive quotients subquotients of the trivial modules $V_i\mathrm{/}{V}_{i-1}$, hence trivial, so $W$ is unipotent; the images of the $V_i$ in $V\mathrm{/}W$ form a flag on $V\mathrm{/}W$ with successive quotients quotients of the $V_i\mathrm{/}{V}_{i-1}$, hence again trivial. For direct sums, concatenating flags adapted to the two summands gives a flag on $V\oplus W$ with trivial successive quotients; for the tensor product, if $g$ acts on $V$ and $W$ by unipotent matrices, then in the tensor basis $g$ acts by the Kronecker product of two upper unitriangular matrices, which is upper unitriangular. Finally, in the basis of [F1] the matrix morphism of this representation factors as $G\to U_n\to {\mathrm{G}\mathrm{L}}_n$ by [F3], giving its asserted upper-unitriangular form without any faithfulness or subgroup-representation extension claim.