On my SWB recumbent, heel strike is an issue.

My personal experience has been that eventually avoiding contact with the wheel becomes second nature and before long you don't even have to think about it. I assume the same would hold true on a road bike with regards to toe strike.

When I use 27x1-1/4" tires, my Peugeot UO-8 has a few mm of overlap, even w/ the 165mm cranks I installed after my son complained about the overlap w/ the 170mm cranks my wife had happily used for years. With my current 700Cx28 front tire, my toes just clear the tire. The only time it was ever an issue was when I was making a sharp turn at very low speed, starting up in a parking lot. A bit of rub, but no harm -- no foul. It is never an issue above walking speed, because one cannot turn the wheel very far when moving fast.

I concur strongly with the advice to corner with the outer foot down and weighted -- this technique greatly improved my confidence and ability on descents.

At one point the consumer product safety commission, which brought us lawyer lips, smoothed front derailleur cages, curved QR levers, etc., got hot and bothered by toe-tire overlap, but I don't know the outcome. If you want to see some serious overlap, look at the picture of Frank Berto trying to shift a 1930s Campagnolo Cambio Corsa transmission in "The Dancing Chain." Since these gear changes included a backpedaling phase*, and since Frank was going slowly to figure out the drill, toe overlap did complicate the learning process.

___
*When your tool is a hammer, every problem looks like a nail. Tullio Campagnolo had invented the quick release and harnessed his patent to build a freewheel based bike transmission in which one loosened the rear axle, then backpedaled to change cogs, then re-secured the axle and pedaled forward in the new gear, with the wheelbase appropriately and automatically adjusted to maintain proper chain tension w/o the common spring-and-tension-idler solution used by almost everyone else.

There's the solution. Wanna buy a 16" wheel?