Unless that reasoning is sufficiently good that the justices will not be embarrassed to sign on, Roberts won't be able to convince them to agree. That social factor is, in fact, a useful limit on the amount of unilateral power that Roberts has as a judge. (On the power of the court as a whole, well, that's a different story...)
Furthermore the ability to write that reasoning is in itself the most important power of the Supreme Court. Because common law says that the reasoning in those cases can then be cited by any lawyer in any federal court in the country as binding precedent. And said precedent will be influential only to the extent that it makes sense, and the lawyer in question can supply an airtight argument that it applies to the case at hand. The more clearly the decision is written, the easier that lawyer's case will become, and the more power that the precedent is likely to have.
Therefore the reasoning that the justices produce for the outcomes that they want matters. A lot.
Furthermore the ability to write that reasoning is in itself the most important power of the Supreme Court. Because common law says that the reasoning in those cases can then be cited by any lawyer in any federal court in the country as binding precedent. And said precedent will be influential only to the extent that it makes sense, and the lawyer in question can supply an airtight argument that it applies to the case at hand. The more clearly the decision is written, the easier that lawyer's case will become, and the more power that the precedent is likely to have.
Therefore the reasoning that the justices produce for the outcomes that they want matters. A lot.