My Route 53 hosted zone has an SOA serial number set to 1, and it does not update even after making changes to the zone. As a result, new TXT records are not recognized by external systems

The situation you're describing with the SOA (Start of Authority) serial number remaining at 1 even after making changes to your Route 53 hosted zone is actually normal behavior. Route 53 does not automatically increment the SOA serial number when you make changes to your DNS records.

The SOA serial number in Route 53 is primarily used for secondary DNS services that support zone transfers. Since Route 53 doesn't support zone transfers, it doesn't automatically update this value. The static SOA serial number does not affect the propagation or recognition of your DNS changes, including new TXT records.

If external systems are not recognizing your new TXT records, it's likely due to other factors rather than the SOA serial number. Here are a few things to check:

1. Ensure that you've correctly added the TXT records to your Route 53 hosted zone.

2. Verify that the name servers (NS records) for your domain in the domain registration settings match those in your Route 53 hosted zone. Mismatched name servers can prevent your DNS changes from being recognized.

3. Check the TTL (Time to Live) values on your records. Lower TTL values will allow changes to propagate more quickly.

4. Be aware that DNS changes can take up to 48 hours to fully propagate across the internet, although it's often much faster.

5. Use DNS lookup tools from multiple locations to check if the new TXT records are visible. Sometimes, local DNS caching can delay the appearance of new records.

If you've verified all these points and are still experiencing issues, you may need to contact AWS Support for further investigation into your specific case.
Sources
NS and SOA records that Amazon Route 53 creates for a public hosted zone - Amazon Route 53
I've updated my Route 53 DNS records several times with no change. | AWS re:Post