In this paper, we investigate unmanned aerial vehicle (UAV) assisted communication systems that require quasi-balanced data rates in uplink (UL) and downlink (DL), as well as users' heterogeneous traffic. To the best of our knowledge, this is the first work to explicitly investigate joint UL-DL optimization for UAV assisted systems under heterogeneous requirements. A hybrid-mode multiple access (HMMA) scheme is proposed toward heterogeneous traffic, where non-orthogonal multiple access (NOMA) targets high average data rate, while orthogonal multiple access (OMA) aims to meet users' instantaneous rate demands by compensating for their rates. HMMA enables a higher degree of freedom in multiple access and achieves a superior minimum average rate among users than the UAV assisted NOMA or OMA schemes. Under HMMA, a joint UL-DL resource allocation algorithm is proposed with a closed-form optimal solution for UL/DL power allocation to achieve quasi-balanced average rates for UL and DL. Furthermore, considering the error propagation in successive interference cancellation (SIC) of NOMA, an enhanced-HMMA scheme is proposed, which demonstrates high robustness against SIC error and a higher minimum average rate than the HMMA scheme.