It is useful to distinguish minis from ‘grid’. People playing 1e-2e could use minis. But it was just as easy to play with or without them. It was 3e that made the grid mandatory, having new rules referring to explicit positioning, for opportunity attacks and so on.
It is an important distinction to make. Minis used on a play surface with or without terrain, required you to measure distances somehow (tape-measures, bits of string, whatever), and positioning had little to go on beyond the base the figure was mounted to. Early D&D and 1e AD&D were written with that in mind, using scale inches, and giving areas in geometric shapes that could be worked out the table. They were adequate rules for that. 2e took very nearly the exact same rules and just dropped the scale conversion, giving everything in feet, and that mostly in 10' increments corresponding to the 1e scale. In both cases, mini's and a play surface were a definite advantage in handling positioning, movement, ranges and areas.
The impact of using a grid was exactly this: You could position minis in squares, instead of by base-to-base contact. And, you could count squares instead of measuring. These both made the use of minis a little quicker and easier. That is the whole impact of using a grid.
The myth that 3.5 or 4e made the grid "mandatory" or that 1e 'required' minis is a total fabrication. In all three cases, though the games presented rules for using minis and a scale of some sort, it was trivial to convert from that scale to the exact same feet as in 2e or 5e or whatever other edition or game you think 'supports' TotM by virtue of lacking such a scale to convert from.
Similarly, rules like the 1e 'parting shot' or backstab, or 3e AoOs, or 4e forced movement, in no way 'require' minis or a surface or a more convenient gridded surface. They're just rules, and converting from the jargon of 'contact' or 'threatened area' to natural language like adjacent or with in reach or whatever you prefer, is trivial. For instance, say it's 2004 and you have an orc charging a fighter who is wielding a longspear. Do you really want to pretend that you need a pair of minis and a grid to determine if the orc can reach the fighter, and will risk get stabbed with the longspear on the way? No. If you can track all the to-the-foot movements, positions and geometric area effects in 5e, you can certainly handle some 5'-granularity movement, straight-line charges, threatened areas, and/or firecubes.
There's simply not a meaningful difference among D&D editions when it comes to adapting them to TotM. They're none of them particularly suited to it. The impression must come from how much better the rules for some of them are when it comes to using minis, creating an expectation that they must trade-off and be somehow 'worse' for TotM. In fact, there's no such tradeoff.
Facilitating TotM is something that games can and have done. D&D just isn't one of those games. You can mod D&D to make TotM easier though, as with Wrecan's SARN-FU, and you can find clones that have better rules for ToTM, like 13th Age.
Where 5e shines, and thus gets away with evangelizing TotM without /mechanically/ supporting it, though, is DM Empowerment. A confidently-empowered DM can sweep mechanics, like to-the-foot granularity or 45-degree cones, that get in the way of TotM under the rug with a simple off the cuff ruling ("Yes, you're close enough"), or facilitate TotM with explicit house rules. That's really where using TotM makes sense in 5e, not because the mechanics are good for, but because running it that way re-enforces the value of and need for the Empowered DM's role in ruling on and resolving all aspects of the game. Players who are accustomed to asking, every round, if they can reach an enemy or how many monsters can catch in their fireball without engulfing their allies, are learning to accept DM judgement as a matter of course. And, the DM is learning to make rulings as a matter of course, for that matter.
