Here is my 2 cents, I have a Meze 99 Classic and Beyer 880/990’s. Do yourself a favor and buy Beyers. They are super-comfortable, I abused them for years and they’re still rock solid. You can backorder almost every part, good quality, the kind that Germany gave its reputation. I like the 770’s the least of the 770, 880, 990 line. In your case I would go for the 880’s, its almost a closed back, non-fatiguing for me, the 250 ohm version if you have some spare cash for dac/amp. Again some more for maybe a DT1990 Pro, they are some of the best bang for your buck headphones in that range with great SQ. (At least here in EU). Would be end game for most normal people. Also much tuning options with changing the pads with Beyers, completely changes how they sound. Mind the different cables between the pro and regular versions, they’re fixed on the 770/880/990 line so you better buy the one that fits your need, personally for PC use I like the 3m curled version of the Pro’s.
The Meze are a ‘fun’ headphone, they fit a bit tighter than the Beyers but my ears still fit. They sit comfortable but not as steady, a bit wobbly, you have to watch out when you lie on the couch. I like them very much as well but I miss some detail sometimes and although they ‘seem’ to do bass well, when I’ve first heard them I thought they did it better than the Beyers, after a while I noticed that it rolls off, for EDM and Hip-Hop it was really starting to bother me once I noticed it (Beeps and boops need that low extension). I’m glad I bought them (was a bit concerned about the ‘dark’ reviews) but I use them mainly for jazz and vocals) and I always go back to my Beyers with great pleasure. They sound nice and detailed with good extension down low and again, super comfortable. I’m not sure the Meze’s will last as long as the Beyer’s, less than a year and already some white’ish protection layer seems to start peeling on the corners of left cup.
I also casually game and there I also like Beyer’s more. More accurate and more information. Probably because they give me more detail and are maybe more designed for a correct presentation in stead of a smooth one.
For amp I think ifi has indeed some great budget options and they are usually also a bit warmer which should be a good match with the Beyers. If you pay some extra for a Asgard 3 with internal DAC (maybe used or b-stock) you’re set to drive any headphone, even a 600 ohm Beyer. A dongle dac from audioquest other brand is also a budget option. To drive the Meze’s you indeed don’t need an amp, they run fine just from phone BUT they sound a bit muffled (at least once you’ve heard them do better). Let’s say they certainly appreciate amplification.
Again, just my 2 cents, you know your situation/usage the best. Maybe the Neo’s dig a bit deeper down low but its a problem for EDM/HipHop on the classics in my book. And again, build quality, comfort, just super on the Beyers. The wood on the meze’s is also very cool though… aargh, choices