One place that particularly needs drainage is near the front of my house, in a planting bed. My driveway prevents me from digging it 10 feet away from the front of the house. There are actually moisture issues on the inside of the crawlspace, which is why I wanted better drainage nearby.

As long as you're not turning the ground into swiss cheese you can drill as close to the foundation as you want. Once you backfill the hole with sand, it's not going to "cave in" and take your house with it.

To avoid sinkhole formation we're going to back fill the holes with sand.

Sand is a better option than gravel because there isn't a large void space to fill causing voids elsewhere. Secondly, unless you have shallow limestone, you're not going to get a true "sinkhole".

Lastly, and the biggest point: installing drains through the clay will cause the clay, especially near the house, to consolidate if the clay is normally consolidated. Now that you have more drainage for water trapped in the clay, the weight of the house will squeeze the water out of the clay voids and cause vertical displacement (Settlement).

As long as you have a consistent spacing of your drains (say every 5 feet) around the perimeter, and you do all of them at one time (don't start half in the summer and wait a year to do the other half next summer), if your house settles, it will be more uniform and will likely not cause problems.