Do a load test on your finished product and prove that Access isn't meant for powering websites.
Write your code so that you can change out the database back end easily. Then when access fails migrate your data to a real db, or MySQL if you have to.
Here are some Microsoft Web server stress tools
For the record, it is possible to send mostly SQL commands to the database and not keep an active connection open, thereby allowing far more than 6 or 7 connections at once, but the fact is that access just isn't meant to do it. So the "it works fine" point is like saying it is fine to clean your floor with sticky tape: it works, but isn't the right tool for the job.
UPDATED ANSWER TO UPDATED QUESTION:
Really the key here is the separation of data access in your code. You should be able to more or less have the data database structure in any number of DBMS. Things can get complicated, but a map of tables should be universal. Then should access not work, decide to use a different database.
Access CAN be used in kinda high traffic sites. With the SQL statement only routines I was able to build an e-commerce site that did a couple million a year in sales, and had 60K visitors a month. It is possible, but maybe not ideal. Those aren't big numbers, but they are the biggest for any site I have been a part of.
Keep access if IT Manager is too busy to maintain another server, or unwilling to spend time configuring one. Ultimately guessing does nothing, and testing tells you everything you need to know. Test and make decisions on the results.