Should I brew at full strength, allow to cool to room temperature before refrigerating, or should I brew at a higher strength and “shock cool” over ice?
The advice I have heard is for stronger tea use more tea. Do not steep longer as it can get bitter.
If you are in a hurry a stronger tea with ice could work but if you want to come from brew temperature to ice temperature it would need to start very strong.
Maybe experiment with double strength and let it cool to like 100 F. You would add as much ice as tea. If all the ice melts then you need to let is cool more next time. If almost none of the ice melts then you need to let is cool less next time.
Tea goes bad like just about every other food and drink, so the faster you cool it the better it will be long term, especially if you're adding sugar to the mix. However, by adding ice directly you water down the tea. Making a super strong tea isn't a good solution as you end up with a higher proportion of bitter compounds, your best bet is to brew the tea normally, add your sugar and flavorings and then cool it as quickly as possible using methods that won't add ice.
My favorite rapid cool method is to stick a load of cutlery in the freezer for an hour, then make the tea and put the ice cold knives, forks and spoons into it for a few minutes. I then remove the cutlery (using tongs, they're hot) and put the tea into the refrigerator to cool the rest of the way. You can also put the pot in an ice bath and stir the tea. I've heard of people using re-usable ice cubes, personally I wouldn't because most of them I've seen are not high temperature plastic, they also tend to leak apparently.
Ideally, if you have the time, try 'cold brewing'. The name suggest you should brew the tea with cold water, and compensate by allowing for a very long steeping time. Basically you can pour water at room temperature over the tea and then let it sit over night. The result is that you still extract all the flavour and leave the bitterness in the leaves. I've tried this before with a dragon well green tea as well as some Yunnan blacks, and it works very well. Never tried it with tea bags though.
What I do is brew a concentrate. I know the number of teabags I need for a 2 quart pitcher of tea, to suit our taste. I steep them in .5 liters of water. (I don't use sugar but, for someone who does, I would add it immediately after steeping.) I then let the concentrate cool slightly for safety reasons.
I then put ice in my pitcher, the amount depending on how soon I plan to drink it. I then pour the concentrate over the ice and fill the rest of the way with water.
This is how I learned to make iced tea growing up. It certainly is much easier than trying to start with the full amount and then adjust to get the desired results. And using this method, the results are always consistent.
I would think the best (and safest) way to achieve this using the coffeemaker would be to use the correct amount of tea for the whole pot, but only half as much water. When brewed, add sugar if you are going to. Then fill the pot the rest of the way with ice and water. This should give you the right strength and consistent results.
How about just cold brewing your tea ?
Not the fastest way to brew tea, but it will get the job done.
Also, I think that making a strong(er) tea will result in a bitter tea; that even with sugar added will not clear the bitter taste.