Game doesn't launch at all - no error appears

avocado_Huss​  This does look very much like a .NET error, but if you've uninstalled and reinstalled .NET 3.5 specifically, and you did NOT get an error when doing so, that shouldn't be the issue.  So try manually installing DirectX 9 as well, as described here:

Re: Sims 3 launcher won't open | EA Forums - 8260715

If that doesn't help either, please uninstall and reinstall the VC++ 2005 x86 runtime.  Open Windows Settings > Apps, scroll down to the Microsoft Visual C++ Redistributables, click the 2005 version (it won't say x86, but it's the right one as long as it doesn't say x64), and uninstall it.  You can ge a fresh copy here:

https://www.microsoft.com/en-us/download/details.aspx?id=26347

You only need the x86 version, not the others.  Restart after installing it and before trying to play.

If neither of these helps, please look for new errors in the Reliability Monitor.  Click Windows key-R and enter "perfmon /rel" without quotes, and you'll see a chart of errors and updates with a column for each day.  Today is on the right.

Look for an error that happened at exactly the time of your most recent attempt to open Sims 3.  If you find one, double-click it to see more details, then copy that info and paste it into a reply here.  If you don't see a new error, check back in an hour or so—the Reliability Monitor doesn't always update right away.