Dirty fireballs are a hypothesized class of relativistic massive-star explosions with an initial Lorentz factor Γ$_init$ below the Γ$_init$ åisebox-0.5ex 100 required to produce a long-duration gamma-ray burst (LGRB), but which could still produce optical emission resembling LGRB afterglows. Here we present the results of a search for on-axis optical afterglows using the Zwicky Transient Facility (ZTF). Our search yielded seven optical transients that resemble on- axis LGRB afterglows in terms of their red colors (g - r > 0 mag), faint host galaxies (r > 23 mag), rapid fading (dr/dt > 1 mag day$^-1$), and in some cases X-ray and radio emission. Spectroscopy of the transient emission within a few days of discovery established cosmological distances (redshift z = 0.876 to 2.9) for six of the seven events, tripling the number of afterglows with redshift measurements discovered by optical surveys without a γ-ray trigger. A likely associated LGRB (GRB 200524A, GRB 210204A, GRB 210212B, and GRB 210610B) was identified for four events (ZTF 20abbiixp/AT 2020kym, ZTF 21aagwbjr/AT 2021buv, ZTF 21aakruew/AT 2021cwd, and ZTF 21abfmpwn/AT 2021qbd) post facto, while three (ZTF 20aajnksq/AT 2020blt, ZTF 21aaeyldq/AT 2021any, and ZTF 21aayokph/AT 2021lfa) had no detected LGRB counterpart. The simplest explanation for the three ``orphan’’ events is that they were regular LGRBs missed by high-energy satellites owing to detector sensitivity and duty cycle, although it is possible that they were intrinsically subluminous in γ-rays or viewed slightly off-axis. We rule out a scenario in which dirty fireballs have a similar energy per solid angle to LGRBs and are an order of magnitude more common. In addition, we set the first direct constraint on the ratio of the opening angles of the material producing γ-rays and the material producing early optical afterglow emission, finding that they must be comparable.