Charles Helliwell's Answer:
Well Sarah, this sounds like a reasonable plan to me, perhaps with a couple of modifications.
Yes, you've inherited a team where the previous incumbent hid many of the team's deficiencies to put a gloss on their tenure as the supervisor. This is more common practice than perhaps you might expect and it's particularly noticeable amongst those looking for a quick promotion or an escape from a hopeless position.
Introducing yourself to the team, as their new supervisor is a great start. However, that's all this is; it's a start. Please don't put all your cards on the table at that meeting. Firstly, use the forum to enable the team to express their opinions and viewpoints on the department; what works well and what doesn't and then allow them to make suggestions on where improvements might be made.
In recognising that they all have opinions and encouraging them to share these, you will find it much easier to present this back to them at a follow up meeting in which you have incorporated your ideas. That way, it looks much more like a collaborative effort than an imposed solution, and you'll have the team, or at least those who wish to remain as part of the team, working with you as opposed to against you.
Once this is done and you have established a consensus of agreement from your team, implement your plan and watch to see who emerges from it as potential team leaders.
Don't assume that the team member with fifteen years of service, who felt that they should have been given your job, will emerge as a leader. They may, or may not. That depends on them and how engaged and supportive they choose to be of a plan in which you will all have a share.