Casting Higher Level Spells from Scroll

I have been wrestling with the case of a multi classed caster trying to cast higher level spell from scroll.  For an example, assume a 4th level wizard / 1st level cleric and a Fireball scroll (3rd level).

TL;DR: Must be able to cast the appropriate level spell or make ability check.

From the Spell Scroll's description on page 200 of the DMG:

If the spell is on your class's spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + spell's level. On a failed check, the spell disappears from the scroll with no other effect. 

In the example case, the spell is on the casters spell list, check.  It is higher than she can normally cast, highest memorizable spell being 2nd level.  Although, the character has a 3rd level spell slot and can use it to upcast spells.

From the PHB chapter on Spellcasting, page 201:

Some spells, such as magic missile and cure wounds, have more powerful effects when cast at a higher level, as detailed in the spell's description.

This seems to state that my example caster can cast spells at 3rd level using that slot and can benefit from using that higher level spell.

I net this out as my example caster can cast 3rd level spells from scrolls that are on any of his lists, normally.  That is no check required.

Is my logic sound, have I missed something obvious?

EDIT Addition Follows:

I posted this question to the D&D 5E facebook group and had definitive responses of yes and no.  several confident answers in both directions.

I had the rules for ritual casting convoluted with scrolls in my head.  I managed to find the rule the ritual casting rule that I had gumming up my works, PHB 169:

The spell must be on the spell list for the class you chose, the spell's level can be no higher than half your level (rounded up), and it must have the ritual tag.

RAW allows casting of higher level ritual spells, based on character level.  It doesn't make that distinction for Scrolls.

Adding the clause "as that class" to RAW clarifies the meaning, at least in my head.  So the rule could have read:

If the spell is on your class's spell list but of a higher level than you can normally cast as that class, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + spell's level. On a failed check, the spell disappears from the scroll with no other effect.

This eliminates "free" up casting from scroll, such that a 19th level cleric, 1st level wizard can't go tossing around high level wizard spells with impunity. 

EDIT 2:

On the other hand, allowing a spell from scroll to be cast by a character who can handle that level, just doesn't have the skills in that class can open up campaign possibilities. So this alternative clarification might be better:

If the spell is on your class's spell list but of a higher level than you can normally cast in any class, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + spell's level. On a failed check, the spell disappears from the scroll with no other effect.

As a default, I am favoring this wording.

Bottom line is this is really murky in the rules, people are convinced on both sides, so I thikn it's best left to the DM and the situation at hand.