In IEEE 802.11bc, the broadcast mode on wireless local area networks (WLANs), data rate control that is based on acknowledgement (ACK) mechanism similar to the one in the current IEEE 802.11 WLANs is not applicable because ACK mechanism is not implemented. This paper addresses this challenge by proposing ACK-less data rate adaptation methods by capturing non-broadcast uplink frames of STAs. In IEEE 802.11bc, an use case is assumed, where a part of STAs in the broadcast recipients is also associated with non-broadcast APs, and such STAs periodically transmit uplink frames including ACK frames. The proposed method is based on the idea that by overhearing such uplink frames, the broadcast AP surveys channel conditions at partial STAs, thereby setting appropriate data rates for the STAs. Furthermore, in order to avoid reception failures in a large portion of STAs, this paper proposes deep reinforcement learning (DRL)-based data rate adaptation framework that uses a sim-to-real approach. Therein, information of reception success/failure at broadcast recipient STAs, that could not be notified to the broadcast AP in real deployments, are made available by simulations beforehand, thereby forming data rate adaptation strategies. Numerical results show that utilizing overheard uplink frames of recipients makes it feasible to manage data rates in ACK-less broadcast WLANs, and using the sim-to-real DRL framework can decrease reception failures.