I would argue that your current approach is not a feasible approach. New versions of browsers are released with zero consideration for Selenium (or any other drivers). As soon as a new browser update is released, there is a reasonably high chance that there will be NO existing driver that works with that version. It often takes days for Selenium teams to release updated drivers to match the newest version of a browser.
And since you are automatically updating your browsers, then you are possibly automatically breaking your Selenium tests until a new driver version is released, or until you downgrade the browsers.
Now, you may be fine with that, and are okay with disabling a browser's tests until the most current Selenium drivers work with the most current browser version. If that is the case, then here are some solutions:
1) If you are using C#, store your drivers in the test solution as a Nuget package, or in a dependencies folder. Then, have the automation reference that driver no matter where it is running. When you need to update the driver, you literally only need to update it in one place, and check in the changes. All client machines will, through your CI process, pull down the latest code, which includes that new driver.
2) If for some reason you do not want the driver in your project as a Nuget package or a manually-saved dependency, then have your CI handle the update process. Point your automation code to a driver located in some common directory on whatever client machine it is currently running on -> wherever your machine stores the dependencies after downloading them. For example; downloading selenium files via console on a Windows machine will put them somewhere in %APPDATA% "C:\Users\xxxxxx\AppData\Roaming\npm\node_modules". That is where your test solution should look.
Then, in your CI scripts, before running any tests, download the latest driver. The syntax is nearly the same, if not identical between Windows and Linux/Unix Kernels. This assumes you have npm installed.
npm install -g selenium
If you already have latest, then nothing will happen. If you don't the latest driver will be downloaded by your CI script before running tests. Then, your test solution will be pointing to where the driver is stored on the client, and it will automatically be using the newest driver.