Wiener's result has been improved several times, and it is hard to tell how big the private exponent must be to be safe from further progress.
Also, the proposed technique, assuming $d>n^{1/3}$, requires a minimum of $\lceil log_2(n)/3\rceil$ modular multiplications for the sparsest $d$ conceivable (a power of two), compared to say ${7\over6} \cdot log_2(n)$ for classical RSA using sliding window exponentiation.
Thus, when not using the Chinese Remainder Theorem, the technique allows a speedup of a factor of $7\over2$ at best, which is less than the factor of nearly 4 allowed by the CRT; and when combining the technique with the CRT, one of the saving in the CRT (halving the size of the exponents) vanishes, thus the speedup is by a factor like $7\over4$ compared to classical RSA with CRT. That's not a huge speedup.
This shows the technique is risky, for a moderate speedup of the private key function (and a huge slowdown of the public key function compared to low-exponent RSA). If that kind of speed compromise is desirable, likely ECDSA is a better choice.
But this leaves the question unanswered.