[Performance] InferenceSession.run executes nodes which are unnecessary for requested outputs

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### Describe the issue

I expected (and maybe that is the bug?) that the `output_names` argument of [`onnxruntime.InferenceSession.run`](https://onnxruntime.ai/docs/api/python/api_summary.html#onnxruntime.InferenceSession.run) is used to avoid unnecessary computations. However, it appears that the entire graph is executed even if only a subset of outputs is requested.

Avoiding potentially expensive and unnecessary computations would be very nice.

### To reproduce

The following code crashes (deliberately) in the `Cast` operation even though it is unnecessary for the computation of the requested "out_c" output. I attached the resulting ONNX graph as a file. I'm happy to type out a test with pure `onnx` functions rather than using [Spox](https://github.com/Quantco/spox) if required.

```python
from spox import argument, build, Tensor
import numpy as np
import onnxruntime as ort
import spox.opset.ai.onnx.v18 as op

def test_avoid_unecessary_compute():
a = argument(Tensor(float, ("N",)))
b = argument(Tensor(str, ("M",)))

c = op.add(a, a)
d = op.cast(b, to=np.int64)

model_proto = build(inputs={"a": a, "b": b}, outputs={"out_c": c, "out_d": d})

session = ort.InferenceSession(model_proto.SerializeToString())

session.run(["out_c"], {"a": [1.0], "b": ["foo"]}). # <= Crashes here

```
The raised error (unsurprisingly) reads:
```
onnxruntime.capi.onnxruntime_pybind11_state.RuntimeException: [ONNXRuntimeError] : 6 : RUNTIME_EXCEPTION : Non-zero status code returned while running Cast node. Name:'Cast_0' Status Message: stoll: no conversion
```

### Urgency

We would like to use this feature in a future project where we would generate a large graph with different expensive-to-compute outputs. We would like to avoid building multiple models which only slightly differ in their set of outputs.

### Platform

Mac

### OS Version

MacOS 12.3.1

### ONNX Runtime Installation

Released Package

### ONNX Runtime Version or Commit ID

1.15.1

### ONNX Runtime API

Python

### Architecture

ARM64

### Execution Provider

Default CPU

### Execution Provider Library Version

_No response_

### Model File

[test.onnx.zip](https://github.com/microsoft/onnxruntime/files/11871714/test.onnx.zip)

### Is this a quantized model?

No

Start with the Python InferenceSession.run entry point and the linked API documentation, then reproduce the attached test.onnx.zip case using the provided output_names and inputs. Trace how requested outputs are selected and verify that nodes unrelated to out_c are not executed; done means the example returns out_c without raising the Cast conversion error.

Written by the indexing model from the issue text.