The time-line events occurred is important here. Chrysler purchased Briggs Body operation in late 1953 after months of negotiations. Since Packard had sourced almost all their bodies from Briggs since 1946, this left them in an untenable situation. Negotiations went forward to buy the Connor Avenue plant both before and after the Chrysler purchase, but Packard couldn't swing the financing.
When Chrysler informed Packard that they would only supply Packard bodies through the 1954 model year, the escape route became a five-year lease of Connor as stopgap. Up to that point, Connor had only been a body production facility with final car assembly still at EGB.
Concurrent to these events, Powers and Grant presented Nance with financial justification to move complete car production into Connor despite it being only approximately one-third the floor space of EGB. The projected savings and benefits were quickly swamped by the cramped reality of a facility woefully inadequate to the task of complete car assembly. Sourcing of component stampings was never really an issue other than as peripheral to the plant question.
Fast forward to the disastrous Spring 1956 when every misstep was coming to fruition. The idea to utilize the 1956 Lincoln body shell to create the Predictor-styled 1957 Packards was hatched as another stopgap when needed financing for the all-new bodies had been turned down by lenders. Nance approached HFII and Breech with the idea; HFII was receptive, but Breech nixed it.
At best, it might have bought them a year or so, though selling a Packard with a recycled Lincoln body wasn't going to be very affective competition against juggernaut Cadillac and upstart Imperial, both all new for 1957.
Dealer for the other three luxury cars would have had a field day reminding prospective buyers of Packard's hand-me-down Lincoln shell. Even Lincoln dealers would have told anyone mentioning Packard that the same value in body quality could be had by simply purchasing their Premiere or Capri. Best of all, not have to worry about whether their expensive new car would soon become an orphan with no resale value.