As a techie, I have designed, built and shipping numerous while labelled software in various industries. In your case, it depends on your priorities, the software itself and any platforms around it.

If you are looking for example to use a white labelled mobile app to complement your core service or a subset of it, you might even use 2 products for A/B testing with focus group before launching to your consumers.

I have seen the above work, and it gives you insight on how to negotiate better deals, and an indepth overview of your real needs, which might not be the most advertsied features of the provider.

One other thing to note is support. If you depend on a third party, it complicates matters further. The provider roadmap is great to plan ahead, and their support hours and timeframe are something to take note of. You will end up needing them and waiting a couple of hours might be the killer.