What do you think of this idea?

The conventional wisdom to something like this is to go validate it in the real world. There isn't much to the idea itself, but there are dumber services out there that are profitable.

Put up a landing page, try some ads, put on a toothpaste suit and dance on a street corner, anything that gets you emails or, even better, real orders. Ship some toothpaste yourself to early adopters, get a feeling of what works and what doesn't.

You probably need high volume to move the needle by selling toothpaste, so finding a few customers manually should be easy enough, otherwise getting tens of thousands of customers quickly, to support a business, is unlikely to happen.

Just my two cents: I would just go to CVS in the corner and buy a new one...or just don't brush my teeth for the day :P You can think about how much pain or problem you try to solve here. Is it painful enough for people to look for solutions?
If you compare to the electric toothbrush subscription or eyeglass, or shaver, they have been quite successful. But they also to try to innovate by making it really cheap.

I personally do not think it would work so well. Think toothpaste is not a good product for a service like that for mainly three reasons (at least that were the first ones that popped in my mind):

1. It is a quite cheap product. So the shipping cost would probably already eat up like 20-30% of the subscription fee.
2. They do not simply go empty overnight. The amount used every time is very little so you see like one week in advance that you will need a new one soon.
3. You buy it literally everywhere. So I think most people get at least 10 chances to buy a new one "naturally" when they are in a store where they have the chance to remember that it is going empty.

Also, do not think it is a big problem to simply have a backup one lying around somewhere. It is not like they are going bad or a lot of capital is bound.

But why don't people just subscribe to their toothpaste on Amazon? (And why don't I?)

I don't think it would work with just toothpaste.
But if you deliver the whole sets like: toothpaste, liquid soap, detergent, washdish soap etc. it would probably work.
Another idea is to deliver toothbrushes because people often forget to buy a new one in time.

Automated delivery via sensor is interesting. I didn't think there'd be a big market for it at first, but bottomless.com seems to have found a market for it for coffee.

I think toothpaste is probably too infrequent a purchase since a single tube can last > month per person. I would dig into why coffee seems to have found some traction and look for similar categories of goods (my perspective is that it's a good with 1) fast turnover (go through a lot, especially for offices) 2) has b2b demand (e.g. restocking offices as well as homes) 3) beans are perishable (so you only buy a relatively small amount at a time e.g. not stocking up a year of supply), etc.