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Dirty fireballs are a hypothesized class of relativistic massive-star explosions with an initial Lorentz factor $Γ_\mathrm{init}$ below the $Γ_\mathrm{init}\sim100$ 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 galaxy ($r>23$ mag), and rapid fading ($dr/dt>1$ mag/day). Spectroscopy of the transient emission within a few days of discovery established cosmological distances ($z=0.876$ to $z=2.9$) for six events, tripling the number of afterglows with redshift measurements discovered by optical surveys without a $γ$-ray trigger. Upon a retrospective search, four events (ZTF20abbiixp/AT2020kym, ZTF21aagwbjr/AT2021buv, ZTF21aakruew/AT2021cwd, ZTF21abfmpwn/AT2021qbd) turned out to have a likely associated LGRB (GRB200524A, GRB210204A, GRB210212B, GRB210610B), while three did not (ZTF20aajnksq/AT2020blt, ZTF21aaeyldq/AT2021any, ZTF21aayokph/AT2021lfa). Our search revealed no definitive new class of events: the simplest explanation for the apparently "orphan" events is that they were regular LGRBs missed by high-energy satellites due to detector sensitivity and duty cycle, although it is possible that they were intrinsically faint 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.