I’ve just done this job on 2010 Fiesta and put a long socket extension bar through centre of old bushes - drilled some rubber while levering my long bar and old bushes came out very easy!!!
Nice video but I don't think its a good idea to refit the new bushing by drawing it in by the core - you could end up damaging the new bushing. The best approach is to use a cup that is the same size as the outer casing and draw it in that way. There is a bit of a profile on the circumference but doesn't interfere too much with the installation.
Iv done a few of these nows an hours job once you know what your doing I use a reciprocating saw to cut them out then put poly's in no special tool required
Not supposed to grease bushings for installation - they are supposed to be a tight friction fit and using grease could cause them to rotate. Best to use a bit of washing-up liquid as that still helps with installation but will will dry off.