There are actually three questions here. The first is: Is there an easy way to see this fact?, to which the answer is (almost certainly) no. Keep in mind that both Chevalley and Borel came to algebraic groups from a background in Lie groups and were therefore well aware of what worked for a topological group. The third question has the same answer: whether the topological group result applies to the algebraic group case. "Long-winded" answers may be a necessary evil here. Keep in mind too that the applications of the foundational results behind your questions deal with whether certain subgroups of algebraic groups are closed and/or connected. There is some essential interaction between those properties.
Todd has worked out the answer to the second question, concerning topological groups. Here the basic results are fairly old, so probably a similar proof is written down somewhere in the literature. But I'll elaborate on some of the other answers and comments about the essential difference in the algebraic group setting.
Algebraic groups (here assumed affine) are given the Zariski topology; in particular, irreducible sets are the more natural refinement of the topological notion of connected. In fact, Chevalley's original arguments about commutator groups and such just used the term "irreducible". It's true that for an algebraic group, which has only finitely many irreducible components (all disjoint), the notions "irreducible" and "connected" coincide. But this can cause confusion at times. In any case, the Zariski product topology isn't the usual one, so all topological arguments involving products and continuous maps have to be rethought in algebraic geometry.
In his 1951 second volume in French on Theorie des groupes de Lie, Chevalley tried out a framework for algebraic groups which proved later to be inadequate in prime characteristic especially (so he changed gears). But he did rethink all the foundational material related to connectedness, which led classically to connectedness of familiar linear groups. His II.7 contains the prototype, hard to read now, of the basic argument.
The notes by Bass of Borel's 1969 Columbia lectures Linear Algebraid Groups adopted more modern algebraic geometry but avoided most scheme language due to time constraints. Chevalley's "long-winded" argument is recast here in I.2.2 and I.2.3 (where part (a) is used to get part (b)). These ideas are crucial for several applications. Coming this early in the theory and used for example to treat solvability, the proofs necessarily rely on first principles. (Borel's expanded second edition, Springer GTM 126, leaves this material unchanged.)
No substantive changes are made in my 1975 book GTM 9 and in Springer's 1981 Birkhauser text. In my book, see 7.5 and 17.2, while in Springer's book see 2.2.6-2.2.8.