Every other caster class once they reach levels 19 & 20 gets a second sixth and seventh-level spell slot. I understand warlocks have so few pact magic slots (2 for the vast majority of their career) to make up for regaining them after a short rest, and that the mystic arcanum/pact magic separation exists so that there aren’t just infinite ninth level spells for a level 20 warlock, but it seems kind of underpowered that they only ever get to know one sixth/seventh-level spell (through the mystic arcanum feature) and can only ever cast it once per day. Can someone explain to me the balance/reasoning behind this?
From a balance perspective, you've hit the nail on the head with your question.
Casting multiple level 9 spells per day up to how many times a Warlock can take a short rest is more powerful than any other spellcaster in 5e.
Warlocks aren't considered a traditional "full caster" like a bard, cleric, druid, sorcerer, and wizard. Nor are they considered a "half caster" like a paladin and ranger. (Half caster and Full caster are not game terms, but colloquially used partly as reference to the levels counted for spell slots in the multiclass section of the PHB)
Pact Magic is something else entirely. Some features can be more powerful than other spellcasters (multiple short-rests per day) and others can be very restrictive (2/3/4 spell slots).
Having the Mystic Arcanum allows a Warlock access to a limited number of powerful spells without the balance issue that might arise from the Pact Magic feature.
Warlocks also have another access to spells that other casters do not, Eldritch Invocations.
Some invocations allow Warlocks to cast spells like invisibility, alter self or hold monster at will without spending a spell slot. In the right hands, and the right campaign, these can often be more useful than a few extra lvl 6-9 spells.