Based on my experience asking customer for reviews using an external authority like TrustPilot have a higher redemption than just sending out a review request using your own brand. For the best results (again in terms of redemption) the review request should be sent out using a sender email address from your organization, and be personalized, but it should be clear at all stages it's an external authority who is collecting it and giving the customer the guarantee it's managed objectively.
Said that, I don't particularly like TrustPilot, I don't know if they changed things, but when they approached me:
- there was no option to import existing reviews
- you was not the owner of the reviews and you could not download them (to keep them if you decide to cancel the contract with TrustPilot)
- there was an upfront fee while competitors had not,
- the sales guy was pretending their widget would have magically made star rating appear on SERP, which is not true.
Once you sign with them, and they set up your business page on their website you get a do-follow backlink from a DA69 website, which is nice, but that's the only SEO benefit you get.
The SERP star rating depends on the schema org markup, if you markup your pages appropriately, it may show up, TrustPilot reviews has no influence on google decision to show star rating for your pages in SERP.
Different story for seller ratings in AdWords, in this case TrustPilot or any other review platform who has a partnership with google will influence your seller rating, and star rating will appear if your adwords account and review match AdWords requirements for seller rating.
As for Conversion Rate, in my experience, showing TrustPilot badge improved CR between 1.5% and 2.5% (depending on website), tests where run changing nothing else, just adding the badge and the review widget, quite far from what they promise.
So I would suggest to choose a review platform and use it in place of an inhouse solution, for the purpose of maximizing redemption. Again, in my experience, using an external provider improved redemption by 3X factor.
But I would look elsewhere for some other platform than TrustPilot.