The best replacement for punt is kick.

The best alternatives to punt are rated as following:
★★★★★  kick
★★★★  drop kick
★★★★  field goal
★★★  bunt
★★  kick off
★★  onside kick
★★  skiff
★  kickoff
  not to punt
  gondola
  place kick

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Speaking about punt, one probably shouldn't forget about punt.

Other search terms related to punt and their ratings are:
★★★★★ punt
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Google Trends

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punt
kick
drop kick