Hard problem to show almost sure boundedness
First observe that $$ S_n(\theta)=2\max_{i\le n} \frac{|e_i|}{\sqrt{1+\theta^2}}\le 2\tau. $$ By assumption there must exist $i$ such that $|t_i|>\tau$. Now for any $n\ge i$ and $\gamma$, for sure we have the lower bound $$ S_n(\gamma)\ge \frac{2|(\theta-\gamma)t_i+e_i|}{\sqrt{1+\gamma^2}}\underset{\gamma\to\pm\infty}{\longrightarrow} 2|t_i|. $$ Thus, for $\gamma$ large enough (uniformly in $n$), we have $S_n(\gamma)>S_n(\theta)$, which immediately implies that $(\tilde{\theta_n})$ must be bounded almost surely.