There was apparently also turbo lift failure below Deck 3 in "Let That Be Your Last Battlefield", forcing Lokai & Bele to make their way down to Deck 5 (at least) by using (unseen) stairways.
And this, I'd think, would be a "failure" engineered by the Cheronites - or more accurately by Bele, to hinder Lokai's attempt at escaping - as they had amply demonstrated their ability to jam and pervert the starship's systems.
In the case of evacuation, they would definately need to have routes that not only avoided turbolifts, but also provided several alternatives in case of hull breaches, decompression, fires etc. Generations shows us just such an example, where Geordi is forced to use the Jefferies Tubes to make his way to the saucer section. In First Contact, the entire remaining crew are able to evacuate the Enterprise without having to fight their way through Borg-controlled decks.
The tri-ladders of the TOS set probably help with this a lot; going up or down one deck should help clear obstacles such as horizontal turboshafts or local breaches easily enough.
The tri-ladders pose the problem of not having a visible means of being sealed either vertically or horizontally, though. Although there could well be horizontal isolation doors just above and below the sections we see, since the between-decks height on the TOS ship might be considerable. And of course there could be air-containing forcefields, as this technology exists in TOS not long after the TOS events, but those probably don't make for a good "fall-back" safety system as they require at least some power.