Motability is a leasing company for disabled people - it is also a charity - but it is not funded by taxpayers. Motability makes profits and it is reasonable to assume that it pays taxes on those profits.
Motability's business model is that for most users of the scheme, it provides brand new cars which are used for 3 years and then returned to Motability and resold. For the term of the lease, most users surrender the mobility component of their PIP award.
There may be some cheaper cars that may be leased for less than the full allowance, but none of the cars I have ever looked at fall in to this category. This is my 3rd car that I have leased.
When applicants decide to use a Motability car, there is a list of cars to choose from and a clear indication of both the cost of the car and the upfront cost. Yes, users of the scheme will have to pay a lump non-refundable sum when they have a car, if it is bigger than a basic one. For example, I currently lease a biggish automatic saloon car because I need its height to be able to get in and out of it and I have difficulties driving a manual car. In addition to all my PIP mobility component, I have paid £1500 for the privilege of having the use of it (for 3 years, remember) and when it is due for renewal, I will have to find a similar sum for another car. I did look at a electric car which would have cost £9000 upfront - luxury cars cost even more.
Motability leases are only 'free', in that a user pays for the lease out of the Mobility component of PIP or DLA - which incidentally, is deducted at source.
Briefly, 2 other points. Firstly, only people in receipt of Higher rate Mobility component are permitted to use the scheme (so not every PIP /DLA claimant is entitled to a car) and secondly, in the event of the user losing their entitlement to Higher rate Mobility (or dying), the car is surrendered immediately.
Motability cars can be quite expensive and it is a choice to lease from them. Not everyone who is eligible for one can afford to use the scheme.
As for claiming PIP fraudulently, I had to go to Tribunal to receive my original DLA award, and when I was eventually assessed for PIP (over the telephone), the assessor made many false inferences from my responses (she says that she plays the organ at church, so must be able to walk the required distance) which I had to refute line by line - a very daunting task - but I was re-awarded on appeal. I would have gone to Tribunal again, because that is where the judgement is made by a doctor (and others).