Direct-detection on-off keying (DD-OOK) weakly-coupled multicore fiber (WC-MCF) systems impaired by intercore crosstalk (ICXT) and laser phase noise are investigated numerically and experimentally. This is performed for systems with a product between the bit rate and the absolute value of the skew between cores much larger than one. It is shown that the phase noise increases the instantaneous ICXT power fluctuations. The standard deviation of the short-term average ICXT (STAXT) power induced by the phase noise depends on the product between the laser linewidth and the absolute value of the skew between cores (linewidth ×| skew|). When linewidth ×| skew|\gg1, typical of distributed feedback (DFB) lasers (linewidth in the few MHz range) and MCFs with skew in the \mus range, the decrease of the standard deviation of the STAXT power induced by the phase noise is 5 dB per decade of linewidth ×| skew| increase. For linewidth ×| skew|\ll1, typical of external cavity lasers (ECLs) and DFB lasers, and MCFs with skew of a few ns, the standard deviation of the STAXT power remains almost unaffected by the phase noise. Experimental results show that, compared with low linewidth ECLs, 10 Gb/s DD-OOK WC-MCF systems using DFB lasers as optical sources in the interfering cores and skew in the range between 2.4 ns and 5.4 ns, may require an additional ICXT margin up to 8 dB for a given outage probability. The additional ICXT margin and the lower amplitude of the STAXT power fluctuations observed experimentally for DFB lasers suggest that the level of the fluctuations of the STAXT power may be an inadequate system performance indicator. The dependence of the outage probability on the interfering core count is also investigated experimentally. It is shown that, for systems with bit rate ×| skew|\gg1, the outage probability only depends on the total ICXT power and not on the interfering core count.