What temperature was the tap water when you mixed the X-Tol? It should be 64-84 F, I believe. If it was colder, maybe all the power did not go into the solution. AFAIK, going hotter than 84 F with the water is of no detriment.
Soda bottles should work fine, so I do not think that is your problem. The PETE plastic is actually a pretty good material for storing photo chemicals. I store most of my chemicals in PETE plastic juice or tea bottles with rubber seals built into the caps.
When I have used X-Tol, I have used it straight or 1:1, so it goes fairly quickly. I never noticed this problem, perhaps because of this.
But I always filter my powder-mixed developer before use; I do it as I pour it from the bottle into the graduate. I put a coffee filter into a funnel and pour slowly. Perhaps this is another reason I never noticed the floaties as being any worse than any other powdered developer. I don't bother with liquid concentrates.
This being said, I don't think the floaties will harm your film if they happen to end up in your working solution. When I used D-76, I often did it by pouring back and reusing until exhaustion, adding time after every so many rolls. There were plenty of floaties then, and no damage to the films.