Recently, Balliu, Brandt, and Olivetti [FOCS '20] showed the first $ω(\log^* n)$ lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed symmetry breaking problems. As a by-product, we obtain improved lower bounds for the distributed MIS problem in trees. For a parameter $k$ and an orientation of the edges of a graph $G$, we say that a subset $S$ of the nodes of $G$ is a $k$-outdegree dominating set if $S$ is a dominating set of $G$ and if in the induced subgraph $G[S]$, every node in $S$ has outdegree at most $k$. Note that for $k=0$, this definition coincides with the definition of an MIS. For a given $k$, we consider the problem of computing a $k$-outdegree dominating set. We show that, even in regular trees of degree at most $Δ$, in the standard \LOCAL model, there exists a constant $ε>0$ such that for $k\leq Δ^ε$, for the problem of computing a $k$-outdegree dominating set, any randomized algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log\log n}\})$ rounds and any deterministic algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log n}\})$ rounds. The proof of our lower bounds is based on the recently highly successful round elimination technique. We provide a novel way to do simplifications for round elimination, which we expect to be of independent interest. Our new proof is considerably simpler than the lower bound proof in [FOCS '20]. In particular, our round elimination proof uses a family of problems that can be described by only a constant number of labels. The existence of such a proof for the MIS problem was believed impossible by the authors of [FOCS '20].