Could not get Lambda functions to get added to RestApi Resource

[Flashminat0]

using Amazon.CDK.AWS.APIGateway;
using Amazon.CDK.AWS.Lambda;
using Constructs;

namespace LambdaWebApi.Dev;

public struct InputFunction
{
    public string FunctionName { get; set; }
    public Function Function { get; set; }
}

public class ApiGatewayStack
{
    public void RegisterApiGateway(Construct scope, InputFunction[] lambdaFunctions)
    {
        var lambdaAPi = new LambdaRestApi(scope, lambdaFunctions[0].FunctionName, new LambdaRestApiProps()
        {
            Handler = lambdaFunctions[0].Function, // Health API function
            Deploy = true,
            RestApiName = lambdaFunctions[0].FunctionName,
        });
        
        
        // Search API and Docs API functions
       var otherFunctions = lambdaFunctions[1..];
        
        foreach (var lambdaFunction in otherFunctions)
        {
            var lambdaIntegration = new LambdaIntegration(lambdaFunction.Function);
            var resource = lambdaAPi.Root.AddResource(lambdaFunction.FunctionName);
            resource.AddMethod("GET", lambdaIntegration);
        }
    }
}

In this code the health API function works with the URL. The others gives 404 errors.

but to make sure the apis are working I tried this

using Amazon.CDK.AWS.APIGateway;
using Amazon.CDK.AWS.Lambda;
using Constructs;

namespace LambdaWebApi.Dev;

public struct InputFunction
{
    public string FunctionName { get; set; }
    public Function Function { get; set; }
}

public class ApiGatewayStack
{
    public void RegisterApiGateway(Construct scope, InputFunction[] lambdaFunctions)
    {
        foreach (var lambdaFunction in lambdaFunctions)
        {
            var lambdaAPi = new LambdaRestApi(scope, lambdaFunction.FunctionName, new LambdaRestApiProps()
            {
                Handler = lambdaFunction.Function,
                Deploy = true,
                RestApiName = lambdaFunction.FunctionName,
            });
        }
    }
}

and it works on different URLs from staged API gateways

what am I doing wrong here. (dotnet 8 all apis are ASP .NET Core web apis from template)

[Flashminat0]

I also tried the RESTAPI method and it does not work as well ( all are 404 )

using Amazon.CDK.AWS.APIGateway;
using Amazon.CDK.AWS.Lambda;
using Constructs;

namespace LambdaWebApi.Dev;

public struct InputFunction
{
    public string FunctionName { get; set; }
    public Function Function { get; set; }
}

public class ApiGatewayStack
{
    public void RegisterApiGateway(Construct scope, InputFunction[] lambdaFunctions)
    {
        var api = new RestApi(scope, "MyApi", new RestApiProps
        {
            RestApiName = "MyApi",
            Deploy = true
        });

        foreach (var lambdaFunction in lambdaFunctions)
        {
            var lambdaIntegration = new LambdaIntegration(lambdaFunction.Function);
            var resource = api.Root.AddResource(lambdaFunction.FunctionName);
            resource.AddMethod("GET", lambdaIntegration);
        }
    }
}