Tile – How to remove Tile levelling clips (T-type) that have partially broken off

The easiest way to remove things is usually in the inverse order they were put in.

If you intend to put down new flooring, you must remove the tile anyway, so remove the tile first. I'll admit that the scariest thing to do might be the tear out. It forces you to accept that you are doing this thing, especially if you have never done something of this magnitude. It will make a mess of things for a wee bit.

I would suggest removing tile with a hammer and wide cold chisel. Break up at least a few tiles by hand. If you have a compressor available, a great tool here would be a flat air powered chisel. This tool would rapidly allow you to peel up the old tile. A bit noisy, but fast. If not, then the hammer and hand held chisel (don't use a wood chisel unless you have borrowed the wood chisel from someone you don't like) will do. Once a few tiles are up, you may find that a large ice scraper, used for breaking the ice from your sidewalk, will be useful to peel up the rest of the tile.

Once the tile has been removed, then remove the baseboard for later replacement.

Demolition can be the most fun part of a project. Try to keep the dust out of the remainder of the house, and use a dust mask.

I wouldn't be too concerned unless there's mortar that is raised above the level of the grout - which you've already said there isn't.

Scraping the higher sections to give a reasonable depth of grout would be as far as I'd go.

A lot of adhesives these days are sold as tile cement and grout, though they do tend to be coarser than "pure" grout, so they are basically the same material. However, a pure grout does give a smoother, "cleaner" line.

Floor tile grout also tends to be coarser - I suspect to be able to cope with the higher degree of wear and tear it will receive compared to wall tile grout.