Flooring – What to use to fill holes in floor tile in kitchen

A few possibilities...

Probably the cheapest (though not necessarily going to work in your setting) is a 3/4" ship auger drill bit through the plate of the wall behind. A slack jawed yokel would just set the bit at an angle through the drywall and go for it, but a sensible individual would open up a small hole in the drywall beforehand to see what they're drilling through. Hitting plumbing/electrical/etc is not in anyone's best interest.

Next would be a diamond grit hole saw. The only bothersome part of this approach is that you'd have to make a 3/4" hole in a scrap of plywood big enough to stand on. That would guide the hole saw and keep it from walking. If you do this, don't center the 3/4" hole saw exactly on your 1/2" hole. Have the rim of the hole saw go through a little bit of empty space in the 1/2" hole, so that it can drain dust and minimize clogging.

Least appealing would be to use a 3/4" hammer drill bit, but if you have other uses for such a thing (and a hammer drill that can actually drive something that big), it might be an option. On low speed and and not on the hammer setting, make the hole bigger. Too much pressure, and you run the risk of cracking the tile. (Not enough pressure and you'll be there for days.) Once you've opened up the tile, the drill bit won't go through the floor very well, but you can probably push your way through with a little more drill speed. From your description, it doesn't sound like you have porcelain tiles, but if you do, avoid the hammer drill bit like the plague -- I assure you that it won't end well.

Two issues:

First, you need to make sure that your floor is rigid enough and planar enough for the tile to be successful. There are deflection tests to determine rigidity, and different ways of checking for planar. The bigger the tiles are, the more important these are.

1/2" plywood is unlikely to be sufficient. 3/4" plywood glued and screwed with backer board on top of it may be sufficient. An isolation membrane (I like Ditra) can help as well, as it lets the floor flex a little without popping tiles or breaking grout lines.

The second issue is floor thickness. If you go 1/4": backer board + ditra + a porcelain tile, that's about 3/4" thick. You need to set things up so the transitions to other rooms are safe; ideally they will be flush, but if not, you'll need a safe transition piece to handled the difference in level.

The installation manual for Ditra is a pretty good reference.