Couple of things here are mixed up.

1) Movement of a site to a new server doesn't mean that one is "starting over".
2) As kobra suggested, there is no need for the URI to be user friendly/readable for search engines.

If there is a desire to continue using a URI rewriter, yes Simple SEO has not been updated to remain usable with more recent versions of PHP/ZC. Either need to use CEON URI rewriter or USU. If having issues with either, take to the applicable forum thread.

As to the description about URI updating for CEON. There is an add-on called CEON URI Mapping manager that is a commercial product to support what you are requesting. There also is a modified version of EasyPopulate version 4 that incorporates the rewriting of the publicly available EasyPopulate v4 and CEON URI Mappings to populate the CEON database with rewritten URIs upon export of the database.

As to incorporating the previously existing URIs to maintain ranking, would need to be able to obtain the previous URIs offered under Simple SEO to then have them rewritten to whatever is newly desired. Otherwise, SEO is all about content...

Good, descriptive content is what is needed for SEO. This may include URIs, but that is just one small aspect of the whole.