I have been using toe clips and straps for 40 years. I tried various cleats and clipless systems (e.g., Look, Speedplay Frogs) and never liked them much. I have wide feet which doesn't help matters.

A few years ago I started using MKS Touring pedals with toe clips (with Shimano touring shoes), but I've started seeing more people using platform pedals with and without pins. Pins seem to make sense but you can gash your shin or tear your tights with them. I recently bought MKS Lambda pedals after watching Mat Ryder ride the Great Divide Mountain Bike Route and a transam from Astoria to New York City on them. 

A big advantage is I can wear any kind of shoe or sandal with Lambdas. Also, during winter I often put some sort of boots over my shoes. No problems with getting my feet in the toe clips. 

(I have to say that one of my pet peeves is riders who use clipless pedals during mass events like the Ride for Your Life yesterday in DC. Over and over, I had to stop while a rider in front of me missed the clip in and had to start over. )

And, by the way, Mat's video series about those tours are fantastic.

I have Shimano M324 pedals on all our bikes. This pedal has SPD on on side and flat on the other so that I can ride with ordinary shoes if necessary. I have Shimano brand MTB shoes.

On the tandem Mary has a M324 pedal on one side but a flat pedal on her dismount side so she does not have to worry about unclipping.

I've been wearing cycling shoes for riding any distance over a couple of km for 35 years or so now, starting with "touring shoes" that, unlike "racing shoes" didn't have the pedal slots on the bottom so were easy to pull out of the toeclip/strap system, especially if you didn't tighten the strap.

Then I needed new shoes and now they all had thicker soles to allow for a recessed SPD cleat, meaning they were rather cramped in the toeclips and the tops of my toes got really sore.  I switched to clip-in pedals, just the basic SPD on both sides, no platform, installed cleats on my shoes, and haven't looked back.  I switched to SPD-SL pedals and compatible shoes for my road bike as the MTB-style shoes I was wearing weren't particularly stiff.  They were reasonably comfortable for walking around, however, though I always brought off-bike footwear on a trip. 

Then, when I returned to doing longer tours after the kids were grown, I started getting "hot foot" issues.  I tried bigger shoes, then special insoles, but I wore out those bigger shoes without ever solving the problem.  A bike mechanic suggested stiffer soles might help, so I invested in some leather Lake shoes with carbon and rubber soles and the problem seems to be solved!  They are very comfortable and the only drawback is that they are as stiff as my road shoes so not great for walking.  (The price is also a drawback; they are the most expensive footwear I've ever bought other than ski boots.)

Like many others, I use the Shimano SPD pedals with one side for clipping in recessed cleats, and a side which is flat for non cleated shoes.
I rarely use the non-cleat side.

Unlike most other cycle tourers, in warmer weather I tour in Shimano cycling sandals. These are fitted with cleats. 

They are cooler, comfortable and saves washing socks. In cold weather I wear  MTBing shoes with recessed cleats which permit walking.

In my younger years, I used toe clips and straps until I discovered they were causing ‘cyclist knee’. Incredibly painful repetitive stress injury of tendons and ligaments on the front of the knee. 

The SPD cleat looseness allows the feet to rotate slightly when pedalling. A lateral movement is more like what happens when walking. 

This simple change of pedal ware made a huge positive difference to my knee health.

Thanks to all for your thoughts. As I guessed, there is a bit of everything. I, myself, use a few different pedal setups on each of my bikes. I think I'll switch my shoe back to a slightly narrower toe box and take it from there.