It's incredible how so many DMs are ignoring the concept of setting up your own enemies. You don't have to use this exact stat block, that is one of the beautiful things about DMing, you're allowed to make an archmage with spell slots that has a list of spells. If you are an experienced enough DM to run a spellcaster with numerous spells and spell slots, you are also experienced enough to homebrew a little and give the archmage spells you want. However, this stat block is good for a DM that wants to just have a spellcaster that they don't have to keep track of as many things for.

Rules as Written, I don't see why not. Rules as Intended, its still classed spellcasting so it might be best to use spellcasting rules despite the lack of spell slots. On the other hand, I believe they did this to be simpler for DMs that don't know how to run spellcasters yet, so it might not be expected of them to know that rule, so maybe RAI is actually that they can?

I guess with RAW being clear and RAI not being clear, you should go with RAW. Though, I don't think it matters all too much, since the person using this statblock is the DM, and they can override and follow rules and a whim if they so wish. They can decide if they want it to work that why or not whether it is RAW or RAI.

I mean, thats kinda what many Wizards are in D&D. Fighting Guards, Commoners, Goblins, Kobolds, any low powered but common infantry would get blasted by fireball or other AOE spells. One Wizard with one Fireball can kill several of these guys in one turn, so spellcasters are uniquely great at taking down small armies, espacilly when they have more spell slots like an archmage would. Not saying I disagree, with it make sense spells wise.

Arcane burst wise, I have no idea how they can arcane burst so often thought. Spells take a while to cast over a sword swing, and even if this is meant to be the "basic magic blast" of many fantasy stories (which in that case why hasn't such an idea been introduced sooner outside of the warlock only Eldritch blast for all classes as a cantrip), why is it so powerful. They are essentially blasting what would essentially be a cantrip based on the very idea of it four times per round, each one doing as much damage as a 17th level Warlock's Eldritch blast. I know you cast 4 Eldritch blasts at that level, but each one does 1d10. You essentially have an Eldritch blasts worth of Eldritch blast (16d10, or 988 average damage, which is more than the 9th level cone of cold assuming you hit two people in hit and one fails their roll [12d8=54] and one succeeds [6d8=27] [54+27=81]).

The damage is so high that it does make me wonder why you would bother casting many of the other more limited spells, especially those who's AOE capability is more sitautional like the line of Lightning Bolt. It feels too strong both damage wise for what it supposed to be and fun wise because why cast the other spells on the list unless you are at an amazing opportunity where you have nearly all players within cone distance and can thus guarantee some damage with an average higher than 4 arcane burst all hitting (if 4 people succeed, it does an average of 4x6d8=108 damage, more if someone fails).