On further review I was mistaken on the insert in the scratch org, although it did work correctly it was because the file that was imported did not have a duplicate record according to the matching rules.  You'll see in the debug log below that the contact list size is '199', this was because one of the records matched according to duplicate rules in Salesforce and so was not imported, but this is where I have found the issue to be originating from.  If the batch does not have any issues with the duplicate rule errors, then it imports successfully.  If there are duplicate errors in the batch, then it has the appearance of a successful import (as shown below) but is ultimately rolled back.

Leads me to my Next Question: Why does having a single record in batch that has a problem with duplicate rules cause only the AsyncRequest__c records from being rolled back for that batch?  The contacts get inserted correctly (all except the one) so there is no issue there, but only the AsyncRequest__c records that were created from the batch get rolled back.  Is there a way for me to catch this rollback in a system.debug of somekind?  I can reliably replicate it, but I can't see it in any debug statement anywhere or in any error.

 
Current Code for my AsyncProcessing class (change is at the end by the insert where I have added more system.debug statements)

public class AsyncProcessing {

	// Simple protection from workflows and triggers
	private static Boolean alreadyProcessed = false;	
	

	public static void handleNPITrigger(List<Contact> ContactList, 
		Map<ID, Contact> newMap, Map<ID, Contact> oldMap, 
		 TriggerOperation operation)
	{
		if(alreadyProcessed) return;
		alreadyProcessed = true;


		List<AsyncRequest__c> newAsyncRequests = new List<AsyncRequest__c>();
		List<String> textChangedIds = new List<ID>();

		System.debug('Total Contacts in Contact List: ' + ContactList.size());
		for(Contact co: ContactList)
		{
			if(operation == TriggerOperation.AFTER_INSERT && !String.isBlank(co.NPI__c) || co.NPI__c!= oldMap.get(co.id).NPI__c && !String.isBlank(co.NPI__c)) {
				textChangedIds.add(co.id); 
			}
			
			if(textChangedIds.size()>99) {
				newAsyncRequests.add(
					new AsyncRequest__c(
						AsyncType__c = 'NPI API Call',
						Params2__c = string.Join(textChangedIds,',')
					)
				);
				System.debug('Current Async Request Size in Loop: '+newAsyncRequests.size());
				textChangedIds.clear();
			}
		}

		if(textChangedIds.size()>0){
			newAsyncRequests.add(
				new AsyncRequest__c(
					AsyncType__c = 'NPI API Call',
					Params2__c = string.Join(textChangedIds,',')
				)
			);
		}
		System.debug('Async Request Size:' + newAsyncRequests.size());
		for( AsyncRequest__c a: newAsyncRequests){
			System.debug(a.Name + a.params2__c);
		}

		Database.SaveResult[] srList = database.insert(newAsyncRequests,false);
		system.debug(srList+'srList+++');
		for (Database.SaveResult sr : srList) {
		if (sr.isSuccess()) {
			// Operation was successful, so get the ID of the record that was processed
			System.debug('Successfully inserted AsyncRequest__c. AsyncRequest__c ID: ' + sr.getId());
		}else {
			for(Database.Error err : sr.getErrors()) {
				System.debug('The following error has occurred.');                    
				System.debug(err.getStatusCode() + ': ' + err.getMessage());
				System.debug('Error Ids: ' + err.getFields());
        	}
		}
		}


		for (AsyncRequest__c ar : newAsyncRequests) {
			System.debug(ar.id);
		}
	}

}

My only Trigger on the AsyncRequest__c object

trigger OnAsyncRequestInsert on AsyncRequest__c (after insert) 
{
    if(Limits.getLimitQueueableJobs() - Limits.getQueueableJobs() > 0)
    try
    {
        QueueableAsyncProcessing.enqueueQueueableAsyncProcessing(null);
    } catch(Exception ex)
    {
        
    }
}

My only other class that performs an update action against the contacts (at the very end)

public without sharing class QueueableAsyncProcessing 
	implements queueable, Database.AllowsCallouts {

 	public void execute(QueueableContext context)
    {
    	if(!AppCustomSetting.appEnabled) return; // On/off switch
    	List<AsyncRequest__c> requests;
    	try
    	{
	    	requests = [Select ID, AsyncType__c, Params2__c
	    		FROM AsyncRequest__c 
	    		where Error__c = false And
	    		CreatedById = :UserInfo.getUserId() 
	    		Limit 1 for update];
    	}
    	catch(Exception ex) { return; }
    	if(requests.size()==0 ) return;
    	
    	AsyncRequest__c currentRequest = requests[0];
    	
    	try
    	{
    		if(currentRequest.AsyncType__c=='NPI API Call') 
				npiAPICall(currentRequest);
    		
    		System.debug(currentRequest);
			currentRequest.Error__c = true;
			currentRequest.Error_Message__c = 'I Ran Succesfully';

			Update currentRequest;

			// delete currentRequest;
			// database.emptyRecycleBin(new List<ID>{currentRequest.id});
    		
    	}
    	catch(Exception ex)
    	{
    		currentRequest.Error__c = true;
    		currentRequest.Error_Message__c = ex.getMessage();
    		update currentRequest;
    	}

    	List<AsyncRequest__c> moreRequests = [Select ID, AsyncType__c, Params2__c 
    		FROM AsyncRequest__c 
    		where Error__c = false 
    		and ID <> :currentRequest.id 
    		and	CreatedById = :UserInfo.getUserId() 
    		Limit 1 ];
    	
    	if(moreRequests.size()==0) return;
    	
		try
		{
			enqueueQueueableAsyncProcessing(context.getJobId());
		}
		catch(Exception ex)
		{
			tryToQueue();
		}
		
    }

	public static void enqueueQueueableAsyncProcessing(ID currentJobId)
	{
		List<AsyncApexJob> jobs = [Select ID, Status, ExtendedStatus from AsyncApexJob 
					where JobType = 'Queueable' And (status='Queued'  Or Status='Holding') 
					and CreatedById = :userinfo.getUserID() and 
					ApexClass.Name='QueueableAsyncProcessing' and ID!= :currentJobId Limit 1 ];
		if(jobs.size()==1) return;	// Already have one queued that isn't this one.
		
		system.enqueueJob(new QueueableAsyncProcessing());
	}

    
    @future
    private static void tryToQueue()
    {
    	if(!AppCustomSetting.appEnabled) return; // On/off switch
    	try
    	{
			if(Limits.getLimitQueueableJobs() - Limits.getQueueableJobs() > 0)
				enqueueQueueableAsyncProcessing(null);
    	}
    	catch(Exception ex)
    	{

    	}
    }

    public void npiAPICall(AsyncRequest__c request)
    {
    	Integer allowedCallouts = 
    	Limits.getLimitCallouts() - Limits.getCallouts();
		if(allowedCallouts<=0) return;
		
		List<ID> idsAfterSplit = request.Params2__c.split(',');
		
		List<Contact> contacts= [SELECT Id, NPI__c, SubSpecialty__c FROM Contact WHERE Id IN :idsAfterSplit LIMIT :allowedCallouts];
			for(Contact c : contacts){
				if(c.npi__c.length() == 10 ){
					Boolean FoundPrimary = False;
					string NPI = 'https://npiregistry.cms.hhs.gov/api/?version=2.1&number='+ c.NPI__c;
			
					HttpRequest req = new HttpRequest();
					req.setEndpoint(NPI);
					req.setMethod('GET');
			
					Http http = new Http();
					HttpResponse res = http.send(req);
					string updatedresponse = res.getBody().replace('"desc":', '"desc_Z":');
					System.debug('resbody:'+updatedresponse);
					JSON2Apex response = (JSON2Apex)JSON.deserialize(updatedresponse, JSON2Apex.Class);
					System.debug(response);
			
					if(response != null && response.result_count == 1){
						for(JSON2Apex.Taxonomies t: response.Results[0].taxonomies){
							if(t.primary) {
								c.SubSpecialty__c = t.desc_Z;
								FoundPrimary = true;
								break;
							}
						}
						if(!FoundPrimary) {
							c.SubSpecialty__c = response.Results[0].taxonomies[0].desc_Z;
						}
						
					}

					else {
						c.SubSpecialty__c = 'Error: No Match for NPI Number';
					}
				}

				else {
					c.SubSpecialty__c = 'Error: NPI Number Incorrect Length';
				}

			}
		database.update(contacts, false);
	}
	
}