Thankyou kindly for your thoughtful reply, and for sharing your efforts,
Wondering whether you have checked on your iPhone that not only is Message Forwarding turned on, but that in the device list for SMS forwarding your Mac is also toggled on. (Both iPhone and Mac need to be signed in to the same Apple Account.)
Set up Messages on iPhone - Apple Support
Eg: says:
"Choose other devices for SMS, MMS, or RCS messaging
Text message forwarding is built into Messages in iCloud. If you don’t want to use Messages in iCloud to save iCloud storage space or to keep messages on only some of your devices, you can still share SMS, MMS, and RCS messages between your Apple devices by using text message forwarding.
When you set up text message forwarding on your iPhone, messages you send or receive appear on your other Apple devices without requiring Messages in iCloud.
Note: Availability varies by region and carrier.
- Go to the Settings app on your iPhone.
- Tap Apps, then tap Messages.
- Tap Text Message Forwarding, then turn on any devices you want to include.
- If your iPhone is set up with Dual SIM, turn on each line. If any devices aren’t appearing in the list, see the Apple Support article Forward text messages from your iPhone to other devices.
Note: If you’re not using two-factor authentication, you need to enter a six-digit activation code that appears on the other device."