(aws-apigateway): `StepFunctionsRestApi` does not pass on `path`

### Describe the bug

The construct `StepFunctionsRestApi` seems to be buggy regarding `path` handling.

### Expected Behavior

When using `StepFunctionsRestApi` with `path: true` I expect my underlying Step Function to receive the path.

### Current Behavior

Passing a path (also tried different depths combinations like: `...prod/users/4` or `.../prod/12`) leads to a `{"message":"Missing Authentication Token"}`

see: `curl -X GET "https://XXXXXXXX.execute-api.eu-central-1.amazonaws.com/prod/123"
{"message":"Missing Authentication Token"}`

### Reproduction Steps

deploy this (almost) minimal example:

```ts
const successState = new Pass(this, "SuccessState");

const stateMachine = new StateMachine(this, "CrudStateMachine", {
definitionBody: DefinitionBody.fromChainable(successState),
timeout: Duration.minutes(5),
stateMachineType: StateMachineType.EXPRESS,
comment: "This state machine returns an employee entry from DynamoDB",
logs: {
destination: new LogGroup(this, "StateMachineLogs", {
retention: RetentionDays.ONE_DAY,
}),
level: LogLevel.ALL,
includeExecutionData: true,
},
});

new StepFunctionsRestApi(this, "StepFunctionsRestApi", {
stateMachine,
path: true,
requestContext: {
httpMethod: true,
},
});
```

and send a similar request to the created API:
`curl -X GET "https://${apigatewayid}.execute-api.eu-central-1.amazonaws.com/prod/123"`

### Possible Solution

_No response_

### Additional Information/Context

I can workaround the problem by creating a proxy integration manually on the API but I guess that is not the idea here.

see

```
const sfnApi = new StepFunctionsRestApi(this, 'StepFunctionsRestApi', {
deploy: true,
stateMachine: sfn,
});

sfnApi.root.addProxy({
defaultIntegration: StepFunctionsIntegration.startExecution(sfn, {
path: true,
querystring: true,
}),
anyMethod: true,
});

sfnApi.root.addResource('testpath').addProxy({
defaultIntegration: StepFunctionsIntegration.startExecution(sfn, {
path: true,
querystring: true,
}),
anyMethod: true,
});
```

Does the current implementation on the vtl `packages/@aws-cdk/aws-apigateway/lib/integrations/stepfunctions.vtl` only work for key value pairs (regarding paths)?

### CDK CLI Version

2.97.0

### Framework Version

2.97.0

### Node.js Version

v18.14.0

### OS

macos 14 (also happened on 13)

### Language

Typescript

### Language Version

^4.8.3

### Other information

_No response_

The entry point is StepFunctionsRestApi and the integration template is packages/@aws-cdk/aws-apigateway/lib/integrations/stepfunctions.vtl. Reproduce the reported GET request, then trace how path: true is rendered; done means a request such as /prod/123 no longer returns Missing Authentication Token and the underlying Step Function receives the path.

Written by the indexing model from the issue text.