Leverage doesn't work that way. True he can't decide where he goes but he does have the ability to refuse to sign a long term deal.
Of course he's also risking the chance of leaving millions on the table in the case of a career ending injury if he chooses to do so.
Even if Gaudreau elects to go for a five year UFA deal, that's not anything out of the ordinary. Kane, Toews, Stamkos, and Kopitar were all on these kind of contracts and it didn't stop those teams from competing or retaining the players, though it did hurt them when UFA time came. And ultimately since it's the team pushing for more term, his annual salary stands to be less if he does not sign for max term, as there is no incentive for not purchasing UFA years. These are all realities of this negotiation. Since he can't sign an offer sheet, he can't inflate his short-term AAV by negotiating with another team. He stands to risk signing a low AAV Ryan Johansen bridge contract if he pushes for a short term, which is a lose for him.
Gaudreau is not pushing for short term just so he can bail for Philly. In fact he's said that he will sign whatever deal his agent thinks is most sensible. The agent may be pushing for short term so that Gaudreau can sign a lucrative retirement UFA deal like Stamkos and Kopitar and Benn just did. The team may be pushing for long term so that he can be cost-controlled throughout his contract like Tarasenko. And chances are, when the contract is signed both sides will have had to compromise on things they would have preferred not to (NTC / NMC / AAV / Term / Signing Bonus) but are ultimately satisfied with the imperfect end result.