So I've been thinking about the same things and have a similar situation. In short, it depends on how often you use your BTC and how paranoid you want to be.
First, let me agree with 99% of Nick ODell's answer. The issue with just keeping a traditional Qt client on your old Mac and only connecting it when you want to spend Bitcoins is keeping the blockchain synchronized. And also because the Qt client has a long startup time as it verifies the existing blockchain. And while the Qt client will allow you to spend coins without being fully synchronized, the transactions you are spending have to be confirmed. Which means you'd need to let the client sync after every time you spend so that you can collect your change.
Which defeats a good part of your strategy since you will have to leave the Mac running, and connected to the network for a long time after each transaction. Most likely you will to leave it running for hours after each transaction.
I also agree with Nick Odell that Armory is the ideal solution for this. You setup a watch-only wallet on your main (online) computer and an offline full wallet on your disconnected Mac. The downside is that it is technically more complex, especially when you are setting up your wallets but also when you are spending BTC.
Another disadvantage is that Armory doesn't seem to have strong Mac support. They were struggling for a while to create a Mac distribution, and although there is now an (unofficial?) Mac version, they don't have a long track record. I found it almost unusable on an old Mac, although much of that was because of the very large amounts of memory in currently uses and Armory would likely be much better behaved for you if you only used it as an offline wallet. (They say this will improve before it leaves beta.)
So, if you are only mildly paranoid, or need a solution that requires less technical knowledge, and you are only spending occasionally, then your approach of keeping client on the network only on the network when you need it may work just fine. Especially if you keep your "petty cash" in an online wallet and just use the Qt client for longer term savings.
Another way to avoid keeping a Qt client connected all of the time is to use paper wallets. Use an online wallet for your petty cash, but move anything you don't need right away to paper wallets. The critical part of this is that you must understand how change is handled.
TL;DR: Yes, a disconnected machine that only runs your Qt client "on demand" will work and is an acceptable way to improve security. But it has a lot of inconveniences and using Armory or an online wallet combined with paper wallets might be a better alternative.