I think volunteers for missions will never be a problem. There will always be people willing to try almost anything.
Think how much easier it would be to put a man on mars if it was a one way mission. So much less fuel and equipment to take with you.
I'm sure you could find thousands of volunteers who would be willing to go even knowing certain death, for the incredible fame and legacy you would leave behind.
When people talk about time travel I always point them towards the direction of Arthur C. Clarke. In the year 2000 he published and released a book by the name of "The Light of Other Days." In this book it takes you through a journey of future Earth among the discovery and mass production of so-called time viewers. Now this I believe, it is a better vision for Time Travel without risk.
Future humanity may not want to risk going through actual physical time travel yes even with robots or holograms because these things could still disrupt space-time. So why not invent a brand new multi media device for viewing the past and the future. What could possibly come next after Holodeck type inventions??? I picture entire rooms could be dedicated to certain areas of the past and people will learn by watching what really happened. As the technology progresses, using augmented reality they could insert a likeness of themselves inside what I like to call history rooms and walk around without anybody knowing. They really aren't there and really aren't interrupting space-time. This will also greatly enhance our intelligence and fill in all the questions we ever had about the past. Using 3D printers we can plug in dimensions of lost historic and priceless artifacts and recreate them in museums.
We may figure out how to physically time travel, but may mark it illegal to do so because of the risk.
This also may explain why there is no record of any time travellers coming from the future.