Might be ok, provided that you are actually dealing with the third parties Insurers. My only concern is when they say you don't have to report to your Insurers. They should know that this is not correct. You still have to advise your Insurers of the accident, even if the third party Insurer arranges the repair of the car and pays relevant additional costs.
Remember that if you use the TP Insurers, you can't use the FOS, if you needed to further a complaint later on. The FOS will only accept complaints from the policyholder about their own Insurance company.
It is up to you. If the TP Insurers are a decent well known company and you are happy to use them to sort out the claim then go ahead. You won't then have the own damage excess on your own policy to worry about. A claim under your own policy, means paying the relevant excess and then claiming back from the TP Insurers. The same is the case with a hire car. Unless your own policy covers this or you are lucky to get a courtesy car, you would have to hire a car, sign an agreement and then hope that the TP Insurers settle the costs. The TP Insurers may arrange a hire car whilst yours is being fixed and pay for it.
Also remember that a claim under your own policy could mean a reduced no claims discount and a higher renewal premium to be paid, until your Insurers have recovered their outlay. This sometimes happens if you have a claim shortly before renewal. The Insurers dock you 2 years no claims and you have to pay the extra, only getting it back, when the Insurers have received the claims payout back from the TP Insurers.