Dwork's use of p-adic analysis in algebraic geometry

The premise of the question is not correct. Dwork's methods (and modern descendants of them) are a major part of modern arithmetic geometry over $p$-adic fields, and of $p$-adic analysis.
You could look at the two volume collection Geometric aspects of Dwork theory to get
some sense of these developments.

Just to say a little bit here: Dwork's approach led to Monsky--Washnitzer cohomology,
which in turn was combined with the theory of crystalline cohomology by Bertheolt to develop the modern theory of rigid cohomology. The $p$-adic analysis of Frobenius actions is also
at the heart of the $p$-adic theory of modular and automorphic forms, and of much of the machinery underlying $p$-adic Hodge theory. The theory of $F$-isocrystals (the $p$-adic analogue of variations of Hodge structure) also grew (at least in part) out of Dworks
ideas.

To get a sense for some of Dwork's techniques, you can look at the Bourbaki report Travaux de Dwork, by Nick Katz, and also at Dwork's difficult masterpiece $p$-adic cycles, which has been a source of insipiration for many arithmetic geometers.

In some sense the $p$-adic theory is more difficult than the $ell$-adic theory, which is
why it took longer to develop. (It is like Hodge theory as compared to singular cohomology.
The latter is already a magnificent theory, but the former is more difficult in the sense that it has more elaborate foundations and a more elaborate formalism, and (related to this) has ties to analysis (elliptic operators, differential equations) that are not present
in the same way in the latter.) [For the experts: I am alluding to $p$-adic Hodge theory,
syntomic cohomology, $p$-adic regulators, Serre--Tate theory, and the like.]