Is there an "Add to cart" button in the front-end extension?

Yes. When using the IS widget, the Add to Cart button is displayed by default on your front-end when the feature "Show products instead of variants" is disabled in the Search options tab.

“algolia_instant_search_product.hogan.liquid”.

At the bottom of the template is this snippet:

[[^ _distinct ]]

      <form id="algolia-add-to-cart-[[ objectID ]]" style="display: none;" action="/cart/add" method="post" enctype="multipart/form-data">

        <input type="hidden" name="id" value="[[ objectID ]]" />

      </form>

      <p class="ais-hit--cart">

        [[# can_order ]]

          <span class="ais-hit--cart-button" data-form-id="algolia-add-to-cart-[[ objectID ]]">

            [[ translations.addToCart ]]

          </span>

        [[/ can_order ]]

        [[^ can_order ]]

          <span class="ais-hit--cart-button ais-hit--cart-button__disabled">

            [[ translations.outOfStock ]]

          </span>

        [[/ can_order ]]

      </p>

    [[/ _distinct ]]

For your active theme, click on "Edit code" and look for a file called "algolia_instant_search_product.hogan.liquid".
This file is responsible for how the search results look.
In this file, look for and comment the code starting with "[[^ _distinct ]]" and ending with "[[/ _distinct ]]".
I've highlighted the relevant portion below: