Because phthalates are not chemically bound to PVC, they can be ingested through food and drink contact and toys children put in their mouths. Also skin absorption from wall and floor coverings may contribute to exposure according to the University of Michigan Formative Children's Environmental Health and Disease Prevention Center.
While removal of the various flooring layers can be done by professionals correctly, covering your existing flooring with ceramic tile, applied to a solid substrate such as concrete board, is a reasonable and lower cost option.
With regard to asbestos, the 'leave it alone' view is supported by an article from the EPAhere. The new ceramic tile will both encapsulate the potential asbestos containing layers as well as put some 'distance' between the vinyl layer and your child's skin.
The condition of the sub-floor and joists should also be noted. Small damaged areas may be repaired from below and not disturb the floor finishes. However, more substantive structural damage, such as water damage below a sink or dishwasher, may require proper floor covering removal to correct the damage.
Good luck with your project,