Why is (javax.servlet.)SingleThreadModel deprecated?

The javadoc says why. SingleThreadModel was designed to be an easy solution to low-load concurrency, but it didn't even manage that:

Note that SingleThreadModel does not solve all thread safety issues. For example, session attributes and static variables can still be accessed by multiple requests on multiple threads at the same time, even when SingleThreadModel servlets are used. It is recommended that a developer take other means to resolve those issues instead of implementing this interface, such as avoiding the usage of an instance variable or synchronizing the block of the code accessing those resources.

If it can't achieve what it was designed for, it should not be used.

It's basically a poor way of handling concurrency. Take the state out of your servlet instead, so that the same servlet can be used by multiple threads concurrently. Keeping state in a "pool" of servlet instances, each of which can have state left over from the previous request etc is pretty horrible.

Yes SingleThreadModel interface is deprecated. Do not use it. In fact you don't need it, instead use local variables instead of object fields since "each thread gets its own copy of local variables in Java. By simply removing the object field and replacing it with a local variable, this particular threading problem is resolved." Reference