You don't have to use the language of measure as the inequality becomes evident when you split $q/\varphi(q)$ into two square roots.

This is true for $\zeta(s)$, so I suspect it is possible to deduce similar results for fixed $\chi$ or for all $\chi$ associated with a fixed modulus, and I guess proving similar results would be more difficult when $\chi$ is a real character.

I doubt whether your choice for $\lambda_d$ is optimal to minimize $S_1$ as it is just a smooth version of Selberg's weight.

Halberstam & Richert's Sieve Theory published in 1974 is another good one if you want to get into some classical additive problems related to primes.

@JesseElliott That may follow from the processes in which logs get plugged into some other logs

I doubt whether one can directly use the $\le$ sign in your first formula as Chebyshev bounds only guarantee $\ll$.

@reuns I don't think Mertens theorem is sufficient to show convergence of that integral. Or otherwise Newman's argument will be unnecessary.

Well, the result from Titchmarsh's book is stronger but requires RH (as the chapter title suggests)