To estimate the covariance parameters of a Gaussian spatial process from noisy observations at a finite number, say $n$, of sites, the methods of solving the likelihood equation or maximizing the likelihood are standard. Evaluating gradients of the log likelihood requires solving linear systems of size $n$ and computing the trace of the associated matrix inverses; and, thus, in many various fields with large $n$, randomized trace estimates have been used for the second task. The purpose of this article is to quantify what is sacrificed when one uses a single (or the same $n_R$, for the version using an average of $n_R$ randomized traces) simulated vector(s) of size $n$ for the all gradient evaluations. We do this mainly in a simple one-dimensional stationary context, with the classical exponential function (thus at most 2 parameters) as covariance of the underlying spatial process, under infill asymptotics, for which consistency and asymptotic distribution results have been shown by Chen, Simpson and Ying (2000). We show that any consistent root of the randomized version of the likelihood equation has the same asymptotic behavior as for the exact version excepted that the asymptotic variances are increased by the factor $1 + 1/n_R$. Moreover, to attack the problem of choosing between multiple, possibly non-consistent, roots, we propose a simple randomized version of the whole likelihood, whose maximizer is proved to be consistent even with $n_R =1$.