Adding a button to the end of a tailstock center is a very workable solution to the problem of not ejecting the center. Another issue is with the tailstock screw.
A longer tailstock screw will eject anything in the taper but will do so at a loss of ram travel. It is a choice by manufacturers. Back in the days when tailstocks were actually longer than they were tall and often times had quite a bit of travel, giving up a little bit of distance of ram travel was more common.
The more modern machines, in an effort to try to extend swings while keeping the costs down, have made tailstocks much shorter in terms of length on the bed and ram length although there are still a few machines that buck this trend. As an example, the TOS I’m currently working on has a tailstock which is quite a bit longer than it is tall, and it will eject anything put in the taper. But it has enough travel length that the loss is insignificant.
So another solution could be to make a new, longer screw if you could live with the tradeoff's.