Are key boxes secure?

The main brand sold here of combination keyboxes can be opened non-destructively - and in a manner that you'll never even know - with a bit of a cut up coke can. The person in this video uses a tool which is basically that but commercially made - a thin semi rigid bit of metal.

Many other brands use the same design. Someone can get your keys, get a copy cut, return them and then enter whenever they want.

(All Master Lock products are awful, as it happens).

Mechanical = decodable. You basically tension up the lock with the handle and try keys to see how they "feel" to figure out potential codes. Like this example, using that specific model of lock!

I don't think there is a single model out there I'd trust to hold lunch money let alone keys to a house.

Depends very much on the door. Can often be retrofitted but may look rather ugly. Ugly vs the cost of a new door, though, is often an easy decision!

Its flat on a table. Nothing being done you can't do on a wall.

There's plenty of stuff which are basically theorethical attacks but this is absolutely something you can do in a normal install. Or just smack it with a hammer.

You need to give them access to keys in an emergency, not for day to day access. Any mechanical key box can be picked. Best to get a good one and have it hidden somewhere. For example in a shed behind the door so you would never know it's there unless told where to look. Or cover it with something.