The failure of the rack is not usually as you describe, but it's possible. Many of the early racks had wear problems that give out befor 150k, but the later ones like yours (post 85) can go a long way. The rack on my '89 was fine at 175k, but had to replace it due to hitting a cement curb. They usually leak first, or the valving goes and you loose power.
If what you say is true, you should be able to see large movement at the wheels while you hold the steering wheel firmly. Jack up the front end and have someone take each wheel and try to turn it while you hold the steering wheel. If the car is off, and there's no power, you should be able to detect movemnet between the rack and pinion gear. Do this throughout the range of turning radius. If it feels more sloppy at the center than the ends, then the rack is worn and needs replacing. The rack to housing radial play in the bushing won't cause what your describing, but rack or pinion gear wear will. If you can live with it and it's not leaking, you may strech some life out of it.
While doing this test, make sure you look at the steering shaft split coupling above the rack input shaft to see that there's no play in it either.
Also, don't take for granted that the new Michelins are balanced that well. In a lot of shops these days they do sloppy jobs and it is easy to have what you describe from a poor tire balance or a weight that fell off.
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'89 245 Sportwagon, '04 V70 2.5T Sportwagon