The GS300 and GS430 both have the starter mounted in the most awkward position underneath the inlet manifold so it is "fun" to remove and some good fitting skills are needed (not for the faint heart).
If your starter motor is in a car with less than say 90K miles with not too many start stop journeys needing the starter it is possible that the solenoidal is just sticking in the housing. However, if you have already replaced the solenoid contacts being "careful to stop them from turning when reassembling" (this is a common mistake with these starters as when bolts are tightened the contacts can move and become misaligned if not held securely in position).
So providing you have got the contacts in square and flat as they should be (no height adjustment is possible) I have had great success without taking the starter motor out (or disassembling the car at all) by squirting a good amount of WD40 from behind the inlet manifold using the tube supplied with the can then "chattering" the solenoid a few times on the key. That works in the WD to help the solenoid actuate and key presto the problem was fixed. Without the extra blood, sweat and tears of a second remove and re-install. At the same time it is worth removing and testing the starter motor relay which is easy and under the cover next to the battery.
Give those little tricks a try. Good luck and "mind your knuckles if to have to take the starter out"
