Phased Array Feed (PAF) receivers are at the forefront of modern day radio astronomy. PAFs are currently being developed for spectral line and radio continuum surveys and to search for pulsars and fast radio bursts. Here, we present results of the pilot survey for pulsars and fast radio bursts using the Focal plane L-band Array for the Green Bank Telescope (FLAG) receiver operating in the frequency range of 1.3--1.5 GHz. With a system temperature of $sim$18 K, the receiver provided unprecedented sensitivity to the survey over an instantaneous field of view (FoV) of 0.1 deg$^{2}$. For the survey, we implemented both time and frequency domain search pipelines designed to find pulsars and fast radio bursts that were validated by test pulsar observations. Although no new sources were found, we were able to demonstrate the capability of this instrument from observations of known pulsars. We report an upper limit on the rate of fast radio bursts above a fluence of 0.36~Jy ms to be 1.3 $times$ 10$^6$ events per day per sky. Using population simulations, we show that the FLAG will find a factor of 2--3 more pulsars in same survey duration compared to its single pixel counterpart at the Green Bank Telescope. We also demonstrate that the new phased array receiver, ALPACA for the Arecibo telescope, will be a superior survey instrument and will find pulsars at a higher rate than most contemporary receivers by a factor of 2--10.
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