Hello, I am Keith, one of the experts on Just Answer, and pleased to be able to help you with your question.
Unfortunately it is; here is the Gov UK web site advice on this subject:
'If you make items owned by your business available for temporary private use, either by using them personally or by lending them to someone else, then you’re making a VAT taxable supply of services.'
Thus the imposition of the tax on the private usage of the car is both right and proper.
I am so sorry to have to rain on your parade.
Yes indeed, all is quite correct.
The company is selling you the usage of its vehicles for private purposes and is correctly charging VAT on these supplies. Equally it is perfectly entitled to reclaim any VAT suffered during its business transactions.
Thank you for your support.