VPN software and Apps are certainly totally valid and useful tools in certain circumstances - but you perhaps need a little understanding as to how they operate and the benefits and protections that they are able to provide.
For simplicity, many VPN Apps will, by design, direct all network traffic through the VPN connection to the VPN Gateway. This is known as “no split tunnel”.
In effect, this ensures that none of your internet traffic accidentally “leaks” and can be intercepted - but also prevents your iPad from connecting to any other computers or resources on your local network while the VPN is active. This will prevent some network services, such as Apple Universal Control and Sidecar from working.
Some of the better VPN clients allow you to configure a traffic policy that allows “split tunnelling” - whereby you can allow access to local network services while forcing all internet traffic through the VPN connection. You’ll need to explore what may (or may not) be available with your VPN App and VPN service provider. You may even be able to use a different VPN and more flexible VPN client with the same VPN provider - but this may take a little more work and technical understanding to get working.