I know from experience that 8 and 9 speed Shimano MTB derailleurs work perfectly with 9 and 10 speed Shimano road shifters. I have that set up on two of my wife's bikes and one of my own. I have a couple of different versions of Deore XT, but don't know the model numbers. They generally work fine up to 11-34t cassette with the "smallest" crank we've used so far being a 48,38,28. We also have a 50-39-30 and a 52-39-30. The chain wrap on all of these set ups is fine. I had to max out the "B" screw on all of them to accommodate the large sprocket and on one I had to replace the B screw with a longer screw.
But the shifting - to my standards - is excellent.
I believe newer 10 speed Shimano MTB derailleurs also work, I just haven't used one yet. I understand they do not come with an on-derailleur adjuster, which I use all the time. That isn't a big deal though. I'm pretty sure the long cage versions of them easily accommodate a 34 and above sprocket, so actually might be easier to adjust than the older 8 and 9 speed ones I've used.
The 8 and 9 speed Deore XT and XTR (IOW, fairly high end at the time) in good shape (not trashed) are readily available and not too expensive on EBay. The regular Deore and lower are available used and new for very inexpensive and I am sure they would work excellently.
Low gears are good and fie on anyone who disparages anyone for using whatever gears they like! I love my 30X34 on my gravel bike and my wife loves her 30X34 on her high end road bike and 28X34 on her gravel bike!