Why does dig +trace seem to ignore the DNS glue records

Glue records only ever exist in the parent zone of a domain name.

Hence in the case of your example.org domain name, first find the .org name servers:

% dig +short org. NS
a0.org.afilias-nst.info.
a2.org.afilias-nst.info.
b0.org.afilias-nst.org.
b2.org.afilias-nst.org.
c0.org.afilias-nst.info.
d0.org.afilias-nst.org.

Then, for as many of these as you feel like testing, explicitly ask those name servers for the NS records for your domain:

% dig +norec @a0.org.afilias-nst.info. example.org. NS

You should get back the correct list of NS records in the "AUTHORITY SECTION". For any name servers that have correctly configured glue you should see those glue A (and/or AAAA) records appear in the "ADDITONAL SECTION".

To add to Mit's answer. The best practice is to only use Glue Records to resolve those circular dependencies. See section 2.3 of RFC1912: http://www.faqs.org/rfcs/rfc1912.html

To quote:

Some people get in the bad habit of putting in a glue record whenever
they add an NS record "just to make sure". Having duplicate glue
records in your zone files just makes it harder when a nameserver moves to a new IP address, or is removed. You'll spend hours trying to figure out why random people still see the old IP address for some host, because someone forgot to change or remove a glue record in some other file. Newer BIND versions will ignore these extra glue records in local zone files.

Note that last bit if you're using BIND. Also, I think most DNS clients will ignore extra-domain glue records, which would explain what you're seeing.

Edit to follow up on comment.

With TLD glue records, you'd typically provide them to your registrar, but otherwise the same rules apply. I run domains in .co.uk and .com. I also have name servers in both of those TLDs that are authoritative for all of my domains.

If I run dig ns co.uk and dig ns com, I can see the big heap of servers that are authoritative for those TLDs. If I pick one of those listed .com servers and dig @<server> ns <mydomain>.com it'll return all four name servers for my domain, but only the glue records of the name servers in the same TLD (supplied as ADDITIONALs), which is what you're describing.

If I run dig @<co.uk server> <myotherdomain>.co.uk, I'll again get all four name servers, but this time accompanied by the glue records for the 3 of them in the .co.uk TLD.

This is all as it should be. If all of your name servers are in the .us TLD, then clients looking for your .com domains will be told the domain names of your name servers by the .com servers; then the clients will perform another query on the .us servers to find their IPs. That extra round trip might sound inefficient, but it only has to happen once per TTL period per client. (typically 48 hours for TLD records.)

(Apologies if you already know all of the above). To answer the original question. Glue records are not intended to save DNS lookups. They're necessary to prevent infinite loops.

Edit 2

Sorry for waffling around the issue. I see what you mean, now (Good diagrams, by the way).

My understanding of how clients should behave is that if one has the opportunity to get authoritative RRs, then it should; but in this case, the client is asking the name server for its own A records, which is pointless, since the Glue is those A records.

I'm reaching the limits of my knowledge here, but I can't think of any situation why a client would need to do this. I'm guessing here, but perhaps the client implementation needs to check that the server is definitely authoritative for ns_.host.us, even when that means what looks like a pointless, extra round-trip.

Try a dig +trace www.example.com to see if it does the same thing. I'd be curious to see if it does the same lookup if the original query was for a _.host.us name, too.