OK - first off - you said 6 tests, 2 in each subject area. Are these the full 50 minute tests? If so -far far too much for one day!
If they are the 10 minute tests, I'd have settled for 1 of each plus a discussion of stuff he got stuck on, rather than two of each, so your DH may have expected too much I think.
It is also quite a tall order to expect many 10 year-old boys to self-supervise and manage their own work. I think you were maybe expecting too much to want him to go away and do his own tests without supervision.
I think you need to have a discussion as lovecheese suggests but with both you and DH present. If the outcome is that he does want to give the Grammar School a try, then I think he will need help to structure his practice.
During the school holiday, I would:
sit him down at the start of the day and explain that you and he are going to do an hour of work. Then the rest of the day is his own (or half an hour with you doing the tests straight away and half an hour in the evening with DH going over them). Nothing else will happen or be done on the day until he has done this hour (or half-hour). Do not allow yourself to be negotiated into leaving it till later in the day.
Put the first 10 minute test in front of him (with sharpened pencil and rubber and a drink and all other stuff kids can faff about getting when they are supposed to be working). Explain he has 10 minutes, if he gets stuck, he should move onto next question, if he finishes, he still has to spend the full ten minutes going back and checking the answers or having another go at a question he got stuck on.
Set a kitchen timer for ten minutes. Let him get on with it without help, but still stay in the same room so he doesn't get a chance to get sidetracked. You can be cooking or ironing or whatever, but still stay there.
If you are doing the whole hour with him - go over paper immediately and discuss problems - if not leave for DH in evening.
Repeat process for papers 2 and 3.
If he needs to go to the loo, change out of an itchy shirt or 101 other displacement activities, the "clock" stops while he does this so he knows that wasting time doesn't get him out of doing any of his "hour".
Praise him for his concentration and effort NOT for how well he has done (even if he does do well). Get DH to do the same! This is really really important.
During term time, an hour at a time will be too much after school. You could try one 10 min paper a night ( ie 20 mins work following the same routine as above).
At some point he needs to get some experience of full length papers so he has some feeling for time management etc. Probably doing one of these per weekend is enough.
A good tutor should be able to help support your DS and build his confidence, which should help his motivation, so don't automatically rule a tutor out because his motivation is low at the moment.