How to redo this table

You can purchase edging for tables and shelves which will fit over the edge. You can paint out the legs and sturdy them up by putting screws into the leg and into the bottom of the table. They will go in on an angle.

I would discourage a mosaic top on this particular piece. Mosaic is heavy and uneven, this is a small fragile looking table. It would be fine just as it is for drinks and snacks and you can do mosaic on a heavier piece!

To make it more sturdy you can replace the brackets the legs screw into, some kits even come with legs, if you want to change those out. Screwing into a pressed wood top is not going to last very long if it get knocked into, or has a heavy load of food and drinks on it. For the edge you can take 1" tiles and place them around the edge, then fill in the top with the mosaic. You can also get a narrow wooden laminate trim, if you prefer a wood look. There are a lot of options for a table like that!

How wide are the gaps between the boards on your deck? I am wondering if the narrow legs had screws on the bottom, as Joy suggested, are they going to get stuck/fall through between the boards?

To be stable enough to support glasses full of drinks, the bottoms of the legs should really be flat and larger.