[SalesForce] How to get the alert on screen for “Submit for approval” when the criteria are met

I've made such button in the past and actually I'm not too happy with it:

1. Users have to be trained to click the non-standard button
2. A formula text field with "TODO list" sits right next to the data and is reportable on. And you might be better off referencing this formula field in entry criteria in approval if they're very complex
3. A small embedded VF page is easier to maintain if your colleagues aren't too good with AJAX toolkit

Here goes:

{!REQUIRESCRIPT("/soap/ajax/24.0/connection.js")}

var poId = '{!Purchase_Order__c.Id}';
if(poId == ''){
    poId = window.location.pathname.replace("/","");
}
var results = sforce.connection.query("SELECT PO_Validation__c, Requestor__c, Approved_Amount__c, Total_Amount__c, PO_Item_Count__c, Required_Delivery_Date__c,Total_Exceeds_Approved_Budget__c, Reason_for_Discrepancy__c FROM Purchase_Order__c WHERE Id = '" + poId + "'");

var po = results.records;
if(po.PO_Validation__c != 'Valid'){
    var msg = 'This Purchase Order cannot be sent for approval before following errors are amended: \r\n\r\n';
    if(po.Requestor__c == null){
        msg += '"Requestor" is blank\r\n';
    }
    if(po.Approved_Amount__c == null){
        msg += '"Approved Amount" is blank\r\n';
    }
    if(po.Total_Amount__c == 0){
        msg += '"Total Amount" is 0\r\n';
    }
    if(po.PO_Item_Count__c == 0){
        msg += 'There are no PO Items\r\n';
    }
    if(po.Required_Delivery_Date__c == null){
        msg += '"Required Delivery Date" is blank\r\n';
    }
    if(po.Total_Exceeds_Approved_Budget__c == 'Yes' && po.Reason_for_Discrepancy__c == null){
        msg += 'Total exceeds the budget but "Reason for Discrepancy" is blank\r\n';
    }
    alert(msg);
} else {
    // Part based on standard Submit for approval button
    if ((Modal.confirm && Modal.confirm('Once you submit this record for approval, you might not be able to edit it or recall it from the approval process depending on your settings. Continue?')) || (!Modal.confirm && window.confirm('Once you submit this record for approval, you might not be able to edit it or recall it from the approval process depending on your settings. Continue?'))) 
        navigateToUrl('/p/process/Submit?retURL=%2F' + poId + '&id=' + poId,'DETAIL','submit');
}

Yeah you're correct in that you can't use a User lookup field as the "Specify Initial Submitters" value.

However, there is a somewhat acceptable workaround. You specify that All Internal Users (or whatever is the most appropriate broad category) can submit the object for approval. But then you add criteria to the Initial Entry Criteria which says that the current user must be the same as Responsible__c ($User.Id = Responsible__c).

The downside is that when the user is not allowed to submit for approval, the error message they get is not user-friendly and doesn't say "You must be the Responsible User".

You can address the error message / initial submission flow with a bit more work by creating a custom button that uses the Approval API to submit for approval, and then you get the luxury of being able to give your users very customized messaging and up-front workflows upon approval submission, but that's quite a bit more work.