Self-supervised learning (SSL) for rich speech representations has achieved empirical success in low-resource Automatic Speech Recognition (ASR) and other speech processing tasks, which can mitigate the necessity of a large amount of transcribed speech and thus has driven a growing demand for on-device ASR and other speech processing. However, advanced speech SSL models have become increasingly large, which contradicts the limited on-device resources. This gap could be more severe in multilingual/multitask scenarios requiring simultaneously recognizing multiple languages or executing multiple speech processing tasks. Additionally, strongly overparameterized speech SSL models tend to suffer from overfitting when being finetuned on low-resource speech corpus. This work aims to enhance the practical usage of speech SSL models towards a win-win in both enhanced efficiency and alleviated overfitting via our proposed S$^3$-Router framework, which for the first time discovers that simply discarding no more than 10% of model weights via only finetuning model connections of speech SSL models can achieve better accuracy over standard weight finetuning on downstream speech processing tasks. More importantly, S$^3$-Router can serve as an all-in-one technique to enable (1) a new finetuning scheme, (2) an efficient multilingual/multitask solution, (3) a state-of-the-art pruning technique, and (4) a new tool to quantitatively analyze the learned speech representation. We believe S$^3$-Router has provided a new perspective for practical deployment of speech SSL models. Our codes are available at: https://github.com/GATECH-EIC/S3-Router.