The table pictured looks like 'art furniture' pretty but not structural enough to use.

Yes get those stretchers up off the floor.

Two suggestions to get some strength in the leg to top connection:

1. Add aprons to get some stability and add another stretcher lengthwise in the center. This makes the base a self supporting structure. Be careful about wood movement.

2. Mortise the legs an inch into the top then get 1 x 1 angle iron pieces at least 8" long and reinforce the connection. (8 pieces) Lots of smaller screws will be better than a few big lag screws. For good measure, add short pieces the other way.