I don't see why not. Umpires have a crazy workload and if any bias or illusion of bias can be shown up by the players then why shouldn't they be allowed to umpire home Tests.
Its not just bias, mistakes etc (though that obviously does play a role).
Its the 2 fold problem of contacts and friend and peers amongst the establishment of one of the sides. There is also a certain level of patronage. Its impossible to not be impacted to a degree one way or another. Even if they are the few that can then that perception will exist.
Technology will not be able to cover everything and the umpires are still in control of the judgement calls. eg Player behaviour, intimidatiory bowling, ball tampering, running on the wicket etc
Most importantly. Its to enable umpires to get out before the **** storm. There will always be a certain amount of controversy.
Now lets take it to near the extreme. Lets assume (rightly or wrongly) that Hair was making what he believed to be an impartial and correct decision in the Oval game.
Now imagine if that was a Pakistani umpire doing exactly same in a game in Pakistan. He then had to live in that Country. His life would be very difficut, if not dangerous.
Old Shep had his confidence ruined by his mistakes being repeatedly shown when giving English players out off no-balls.
Ship 'em in and ship 'em out again. There is no reason for an Umpire to have to live on a day to day basis with controversy in their home Country.
Make your decisions and move on. Tomorrow is another day.
If referals were to happen it would have to be like NFL where there would have to be 100% conclusive proof that a mistake was made in order to reverse the decision. 99% isnt enough.
In cricket, its the 50-50 decisions that are subject to bias and are hard to tell whether its the rub of the greeen or intentional.