Sure they can. The roster in these playoffs was completely different than last year's, and yet Tortorella continued to employ the same tactics and systems. He failed at maximizing individual player strengths. He failed to get his coaching staff to employ anything resembling successful tactics on the power play. He continued to shorten the bench while the opposition rolled 4 lines and 3 pairs.
Even if the next coach does everything exactly how Tortorella did it, but improved his power play tactics, the team will have more success.
This roster isn't garbage. There is enough talent on this team to have home ice in the playoffs (under the current format). And they will address some needs this summer.
This is why it is important to get whoever is their guy, hired sooner, rather then later. They need it done before the draft so he can pick his staff, go through the process of hiring them, have the organization on the same page regarding the new philosophy, what roster needs are, etc etc. The draft is where rosters start to form for the next season. The staff needs to be ready to provide their input for their needs during free agency.
Then the coach needs to be in contact with his players to let them know what to expect and what to be prepared for.
A new coaching staff could absolutely have a major impact on the success and/or failure of this roster. A good coach and a good staff will be able to get more out of what they have.