I realize I never answered the first/most basic question "how do I explain that narratively"
Gord-Gygax(Thief acrobat) is fighting a fire giant. The fight started promisingly, with a surprise backstab, but the giant, after bellowing out in pain & shock, simply turned(morale test made), and, eyes lowering to the rogue, devolved into melee combat, the immensely strong, thickly built giant slashing ferociously with his 12' greatsword. Gord-Gygax, by contrast is trying to stay in close while avoiding blows, seeking gaps in the giant's armor, or at least to land thrusts into mail rather than plate. Gord is using his long sword(+2, actually a holy avenger, the DM being the only one to know this fact, pillaged from a tomb long hence), but also his dagger(+3 vs larger than man sized foes) in his off hand (dex 18; 0/-1 mods).
we roll for initiative, 6 for giant, 2 for Thief-Acrobat. The wounded, furious giant lashes out (rolls 11; with HD 11+4 that hits ac -2; Gord has: bracers of defense ac 5, ring of protection +1, dex 18, cloak of protection +3 for AC -3, a miss ). DM says "Gord just manages to duck beneath a mighty slash, the giant having a hard time adjusting to his proximity & height, roll for his sword & dagger attack".
The player does (THACO 16 as L10 master thief-acrobat, and 17 str for +1, magic weapons & off hand mods noted above): 2+3=5, miss, and 8+3=11, miss (giant is ac 3)
DM "scrambling about as he is to avoid a fatal blow, Gord's sword bounces from armor plates, while his dagger thrusts are foiled by mail, thus far at least. "
both continue melee the next round, initiative comes up 2 giant and 4 thief-acrobat
Gords attacks 15 hit, 11 miss, for 5 damage, but the giant remains standing (HP 12/54). DM "Gords blade finds gap & penetrates, though not deeply. The black skinned giant seems to be a bit gray now though, as the effects of the backstab seem to be sinking in...still, the now desperate giant slashes and swipes at you" rolls 15 "and it would be a hit, but lets test your evasion ability" player rolls 91% failing (base 30% +5% dex +4% no armor=39%). "sorry, Gord almost manages to avoid the giant, but he is already beginning to tire from the desperate pace, and likely to run out of luck soon" rolls "remove 16 hp" (Gord has 44/60 now; HL character HP are mostly abstractions, skill, luck, etc, in 1e, until the numbers are low).
Initiative of 6 giant to 2 acrobat results in the giant strike missing (8= ac 1, miss) & sword (12) miss & dagger (12) miss (by 1!)
Round 4 is giant 1 thief 6, Gord attacks 17 hit, nat 1 miss for 6 damage (the player curses). The giant staggers about, at 6 hp, but attacks 7 (missing)...actually skip that the example is taking too long to play out with actual die rolls
if the character losses initiative in melee, claim they were caught out, seeking to land strikes of their own.
The important parts are that a) initiative is rolled every round b) combat rounds move fast in AD&D, c) segments only matter when they matter (basically when spells [or spell devices/powers] are cast). d) evasion is a mild bonus in most cases. It's always potentially far better vs ranged attacks; as I'd ague it applies to any attack the character is aware of & of small enough size, not needing initiative, and more creatures can shoot at a character than can engage in melee anyway.
tumble: evasion is the only ability of real note for the class, tumble: attack maybe is in some games, dependent on pummeling rules used/ruled on...and it's usually of negligible importance. Most play simply is not at L15+. It can be cool, and clutch on those blue moons or right circumstances though.
Initiative can be simple (as I'd suggest) or not. Even adding some complexity (generally on TIED results) or the seemingly difficult casting ideas doesn't have to be that hard.