I would go with not re-signing Alfredsson after using Alfredsson's contract to circumvent the salary cap.
For those unfamiliar, Alfredsson signed what was supposed to be a 3 year 18.5M (6.17M AAV) contract. There was a fourth year tacked on at 1M in order to lower the AAV to 4.875M. The understanding was that Alfredsson was not going to play out the contract.
During the life span of the contract, the Senators went into a re-tool and saw no real cap benefit from the circumvention. They would have been fine with Alfredsson at 6.17M AAV. With the final year of the contract being a lockout shortened year, and Alfredsson feeling okay, he played out the contract with the understanding that he would be overpaid on an extension to compensate him for playing for only 1M. Keep in mind that the Senators were not going to be anywhere near the salary cap, so paying Alfredsson 7M for 1 season would not hurt them in that regard.
When it came time to pay Alfredsson, Melnyk allegedly went back on his word, or Bryan Murray may have fabricated the team's willingness to compensate Alfredsson. This resulted in Alfredsson leaving Ottawa. Keep in mind, the team did not have any cap issues that would prevent this kind of extension. The concerns were purely budgetary.
This set off a chain of events that set the franchise back years both in marketing perception, and on ice success. First, the big selling point as to why Alfredsson should take less money was that Bryan Murray was working on a trade for Bobby Ryan. Re-signing Alfredsson would not allow them to afford Ryan's salary. At the time, that seemed like a bad thing. Looking back, had the team not acquired Bobby Ryan they would have kept Jakob Silfverberg, their first round pick, and they would have avoided a massive 7M+ contract that helped sink their chances of competing under a budget with Karlsson in his prime. Not to mention, Ryan was phenomenal in his first year in Ottawa, at least for the first 40-50 games. With a broken down Alfredsson and no Bobby Ryan, there is a chance that the Senators would have finished near the bottom of the league where they could have gotten one of Ekblad, Bennett, Reinhart, or Draisaitl (Dorion apparently loved Draisaitl). Not to mention, it may have expedited a Spezza trade if the Senators were further out of the playoffs at the deadline, and possibly lead to them getting back a better return than Chiasson and a 2nd.
While "the Senators would have sucked more in 2013-14" might not be a very compelling argument at first, adding Silfverberg, a possible top 4 pick from the 2014 draft, a better return for Spezza, and 7M of cap space long term would put the Senators in a better position to build around Karlsson.
Not to mention, prior to Alfredsson leaving, Melnyk was seen by the fans as the fun cowboy type. Your crazy uncle who knew how to walk the line at family dinners between outrageous and offensive, but was always there when you needed him most. Internally, we now know that there was always tribulation around Melnyk as the owner, but this had yet to leak out to the general public. Alfie-gate was the first instance of the mask slipping. It started a snowball that nearly killed the franchise. Melnyk's personal tainted brand became so far intertwined with the team, that nobody wanted to pay to go to games. The above may have still happened, but Alfredsson not being forced out initially (he probably leaves eventually) may have stopped some of the bleeding and bought the team a few more years of goodwill from the "pizza line" years.