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We introduce a simplified fabrication technique for Josephson junctions and demonstrate superconducting Xmon qubits with $T_1$ relaxation times averaging above 50$~μ$s ($Q>$1.5$\times$ 10$^6$). Current shadow-evaporation techniques for aluminum-based Josephson junctions require a separate lithography step to deposit a patch that makes a galvanic, superconducting connection between the junction electrodes and the circuit wiring layer. The patch connection eliminates parasitic junctions, which otherwise contribute significantly to dielectric loss. In our patch-integrated cross-type (PICT) junction technique, we use one lithography step and one vacuum cycle to evaporate both the junction electrodes and the patch. In a study of more than 3600 junctions, we show an average resistance variation of 3.7$\%$ on a wafer that contains forty 0.5$\times$0.5-cm$^2$ chips, with junction areas ranging between 0.01 and 0.16 $μ$m$^2$. The average on-chip spread in resistance is 2.7$\%$, with 20 chips varying between 1.4 and 2$\%$. For the junction sizes used for transmon qubits, we deduce a wafer-level transition-frequency variation of 1.7-2.5$\%$. We show that 60-70$\%$ of this variation is attributed to junction-area fluctuations, while the rest is caused by tunnel-junction inhomogeneity. Such high frequency predictability is a requirement for scaling-up the number of qubits in a quantum computer.