Now, I'm not going to say that the M-stage can power the K702s adequately as they are a very hungry phone (less so than the k701s in my experience but still pretty hungry). But I will say that if you're wanting to continue using your sr-80s then the gain switch on the m-stage being more accessible than the gain switch(es) on the LD3 could be important. On the M-stage the lo vs. hi gain switch is right there on the front panel.
As you can see on this page of pictures of the LD3 and it's internals:
http://www.littledot.net/forum/viewtopic.php?f=9&t=815 You have to open the case of the LD3 in order to switch the gain.
You'll see "GAIN SW1" and "GAIN SW2" printed in white on the pcb next to the little red boxes with white switches in the 5th, 8th, 9th, 10th pictures on the page if i counted correctly.
I'm not saying you'll have to do this every time you switch from the akgs to the grados but you might have to do it multiple times if you have a little trouble finding the right gain setting so that the AKGs aren't too quiet and the grados aren't too loud. Where the sweet spot is depends on a number of factors, including your source. The LD3 does not use a stepped attenuator so there might be a tiny but perceptible channel imbalance (where the left channel will be a little louder than the right channel or vice versa) at low gain. Which is all the more reason to make sure you get the gain right.
To address your question about the different technologies, I haven't heard the stock LD3 but in my experience solid state is generally a better value at the entry price range if you're not prepared to tube roll. I've liked the m-stage and lovely cube that i've heard (wish i could have compared them to a real lehmann at the time) and I would be leaning towards that, but there are a lot of happy LD3 owners out there, so I would say you can't go wrong either way. If you don't mind replacing tubes eventually, or the prospect of tube rolling sounds exciting to you, then I'd say this draw is a win-win as both are great amps for the price, and the AKG K70Xs that I've heard had good synergy with tube amps that I've heard.