Being a member of the HPC and CSP are two seperate things.
However the CSP provides the insurance and trade union for most HPC Physiotherapists who work in the UK. If you aren't a member of the CSP, you will need public liability and professional indemnity insurance through someone else, which HPC membership doesn't give you.
Membership of the CSP is not, as far as I am aware, open to all HPC registered Physiotherapists. You need to be on their list of recognised courses to join. So you are HPC registered, but not done a recognised course for CSP membership, which is why you were grandparented in, presumably because you had experience which was considered acceptable by the HPC, nothing to do with the CSP.
I think the key issue here is your insurance. Some organisations do ask for MCSP because they now what the course content is, others are happy with HPC membership. Frankly I would look for CSP membership if I employed a Physiotherapist in my private practice