When we play (TotM), we state distances in terms of moves instead of feet. This can get a bit messy with characters with a movement distance larger than 30', but we usually just assume that they move 1.5 moves and round in favor of the moving character when necessary.
Perfectly workable. You loose some of the granularity of feet, but everything's given in increments of 5 or 10 ft anyway, so it's hardly a meaningful loss.
Playing Theatre of the Mind can require a fair amount of trust between players and DM.
Playing D&D, especially 0D&D/1e/5e, requires a considerable amount of trust in the DM, already. The gp should have "In DM We Trust" on one side.
I am sure that the level of ambiguity that Theatre of the Mind can create annoys some players and DMs to no end. However, it can be advantageous to dramatic action. If a player has an whacky and exciting plan involving a chicken, a banister, and a cooking pot, the details about how far away each of those items might be can be easily molded if their location hasn't already been pinned down.
Nod. That shades into the 'story now' theories that spawn huge threads, but yeah, a good DM (or player) can leverage ambiguity.
I find that in Theatre of the Mind its much easier for me as the DM to say "Yes."
I find it's easier to say 'no' to avoid complicating the scene you're trying to hold in your head, too. :shrug: I ran TotM a lot for a good 10 years. (There was a period in the 90s when there was just no where for the groups I gamed with to run that actually had any kind of useable play surface, the last FLGS with table space closed down in late 80s, so the decade was bereft that way, and we were mostly college kids with little space at home.) It definitely pulls the GM (mind, this was mostly Storyteller & Hero, the latter being more 'grid dependent' by design than any ed of D&D ever contemplated being) towards simplified combats with less terrain, fewer options, and fewer enemies (1 being ideal - and, not coincidentally, I think, 5e encounter guidelines are more dependable when the party has numeric superiority).
New players (and sometimes seasoned ones) often need a little coaxing. Instead of asking, "How far away is that chicken, and what's the distance between the chicken and the banister?"
That's yer seasoned player, right there.

You can tell the high-player-skill/CaW types by the way they ask you seemingly stupid questions cagily, then spring an action on you based on them.

Annoying...
They should say, "I have this plan involving a chicken, a banister, and a cooking pot. It goes something like this, . . . . . and the pot with the chicken ends up here! Is that possible?" The first set of questions restricts what might happen later. The second set allows the DM to simply say "yes" to a fun scene and only at that point decide the exact locations of said chicken, banister, and pot.
Agreed, but breaking some players of habits learned in even semi-adversarial campaigns requires some patience. It's nice if you also have the trusting/enthusiastic player, so the cynical ones can see harebrained schemes working as the world subtly shifts in their favor.