Through the lens of average and peak age-of-information (AoI), this paper takes a fresh look into the uplink medium access solutions for mission-critical (MC) communication coexisting with enhanced mobile broadband (eMBB) service. Considering the stochastic packet arrivals from an MC user, we study three access schemes: orthogonal multiple access (OMA) with eMBB preemption (puncturing), non-orthogonal multiple access (NOMA), and rate-splitting multiple access (RSMA), the latter two both with concurrent eMBB transmissions. Puncturing is found to reduce both average AoI and peak AoI (PAoI) violation probability but at the expense of decreased eMBB user rates and increased signaling complexity. Conversely, NOMA and RSMA offer higher eMBB rates but may lead to MC packet loss and AoI degradation. The paper systematically investigates the conditions under which NOMA or RSMA can closely match the average AoI and PAoI violation performance of puncturing while maintaining data rate gains. Closed-form expressions for average AoI and PAoI violation probability are derived, and conditions on the eMBB and MC channel gain difference with respect to the base station are analyzed. Additionally, optimal power and rate splitting factors in RSMA are determined through an exhaustive search to minimize MC outage probability. Notably, our results indicate that with a small loss in the average AoI and PAoI violation probability the eMBB rate in NOMA and RSMA can be approximately five times higher than that achieved through puncturing.