This paper studies low-power random access protocols for timely status update systems with information freshness requirements, measured by age of information (AoI). A fundamental challenge for such networks is to schedule a large number of transmitters to access the wireless channel in a way that achieves a low network-wide AoI at low power consumption. Conventional packet-based random access protocols involve transmitters contending for the channel by sending their entire data packets. When packets are of long duration, the time and energy wasted due to packet collisions is considerable. In contrast, connection-based random access protocols establish connections with the receiver before transmitting data packets. From an information freshness perspective, there should be conditions that favor one approach over the other. Therefore, we conduct a comparative study of the average AoI of packet-based and connection-based random access protocols. Specifically, we consider frame slotted Aloha (FSA) as a representative of packet-based random access and design a request-then-access (RTA) protocol for connection-based random access. Our analyses indicate that the choice between packet-based and connection-based protocols depends mainly on the payload size of update packets and the transmit power budget. In particular, RTA significantly reduces AoI and saves power, especially when the payload size is large. Overall, our investigation offers insights into the practical design of random access protocols for low-power timely status update systems.