Steering shudder

Check the brake disc for warping. If you've crossed streams with hot discs and pads the warppng effect can cause considerable shudder.

Sometimes tyres change to an egg shape instead of being perfectly round when they have done a few km.
This will cause a wobble at speeds. Why this can happen i dont know.

The factory tyres are not thre greatest tyres around & considering they are already at 80,000km id suggest a new set would be on the cards.

If not wanting to renew your tyres try borrowing a set of a mate maybe & see if the problem is still there.

Kev.

I had an FJ40 years ago. Same problem (but it was at 80kph+). Well known brand of tyres, supposedly balanced, were 6mm out of round when checked with a dial indicator while vehicle was jacked up. Tyre dealer took some convincing but replaced them. Problem fixed!
Anything more than 2-3mm out of round will cause havoc at high speed.
You don't need a dial indicator. Jack up vehicle until wheel just clears ground, spin wheel by hand and observe gap between tyre and ground. $200 tyre - expect perfection!!!