it sounds like you do have a small amount of play somewhere & you just havn`t found it (yet)
however,,, most minis have "This" problem anyways, & the older they are & the more abuse they have seen over the years, then the more "This" problem shows it`s ugly head
now, "this" problem is simply-->
Jack your car up on the passengers side, so that front wheel is in the air, turn the steering wheel full lock to the right, & then Have someone "wobble" the left front wheel quite firmly to & for, you sit inside the car & watch the front floor where the left/hand/side steering rack mounting is,,, you will most surely notice the floor flex in & out
this will creat some form of "bump-steer"
every mini has it to differing degrees,,, it became quite bad in our old speedway minis & rally minis where we then made stiffening ribs/braces to weld up around that area to help stop it
at one hillclimb i was even declined entry upon scrutineering because they deemed it unsafe as they figured the steering rack itself was at fault,,, untill i showed then the floor flex
it`s quite common but fairly un-noticed by the masses
it may be as simple as that, but it may be just that you have some play in the system still but have yet to find it
even a pair of "miss-matched" tie-bar rubbers will do it,,, like one is stiffer than the other, is all it takes
a tiny bit of "spongey-ness" in the left hand end rack bush is all it takes
a miss-match of lower inner control arm bushes will also do it,,, again one slightly stiffer than the other
have a far far better look & you will (should) find the problem
it`s not un-common at all,,, but i don`t stand for it,,, i won`t stand for it,,, i will not,,, & never will,,, i find it much better to sit down
