Try doing it with one.
My son has a very good hedge trimmer, the blade is a disc
within a protective frame. I will find the name if you want to get one.

The roots will die if you cut them off, but it takes many years. The trees might well be damaged or destabilised as well. The trees will always be a problem for your lawn, personally I would have the trees removed. I did this in a similar situation and the lawn has recovered nicely, even though I didn't try to remove any roots under it.

Agree, the best plan is to remove the trees. As they will have greatly impoverished the ground around them you will then need to improve it to get it into good heart again by feeding the lawn in the spring.

The soil between the roots is extremely compacted and hard. To remove them is an impossible situation by hand and tools even if the trees are removed. You have a few years of soil rehab ahead of you. I would suggest you have a landscaper take a look at this.

When they were removed from my garden - they were originally planted by the council to improve water seepage off the playing field at the back, cost £400 per tree
as kids from the school set fire to them -
the men told me there was no need to put Stump Remover Treatment on the stumps because they would never grow again.
Might be worth laying artificial grass for a few years, I see
it costs £30 a running yard, about 10 feet wide (I didnt measure the width in the garden centre.) This might be metres too. Good luck.