The Thief archetype Rogue gets the Use Magic Device feature at 13th level:
By 13th level, you have learned enough about the workings of magic that you can improvise the use of items even when they are not intended for you. You ignore all class, race, and level requirements on the use of magic items.
Some of these items require the use of the user's own spellcasting ability and proficiency bonus. The rogue would only have the latter, but would the Magic Initiate feat fix that?
You have a Spellcasting Ability, not a Spellcasting Class Feature
Magic Initiate states (PHB, Chapter 6-Feats) with my emphasis:
Your spellcasting ability for these spells depends on the class you chose: Charisma for bard, sorcerer, or warlock; Wisdom for cleric or druid; or Intelligence for wizard.
The spellcasting ability is directly related to just those spells and are a result of the class you chose when taking the Feat. However, this does not grant you the Spellcasting Class Feature.
Other than the strict reading I provided above, I'm not sure why it would be a big deal to allow the Rogue to have a spellcasting DC for those types of items that require it.
The feat is called Magic Initiate, after all. They have been initiated into Magic of a certain Class. Let the Rogue have a spellcasting ability in that class in general and utilize the associated ability score for magic items that require it.
From the DMG errata about attunement:
If the prerequisite is to be a spellcaster, a creature qualifies if it can cast at least one spell using its traits or features
Having the Magic Initiate feat does fulfill these criteria. Items that require you to be a spellcaster reference your spellcasting ability and related stats without specifying any further restrictions on how you got one.
Make a ranged attack roll against that creature using your spell attack bonus. (DMG 211, wand of paralysis)
There is also no feature as of date that provides you with a general spellcasting ability. All of them are linked to a class and they are called "your wizard spell save DC" or the like. So while it is not explicitly stated, these all assume that you will use a spellcasting ability linked to a class when casting from the item. If the limitation that you can only use that DC for, say, wizard spells would affect items, wizards would also be unable to utilize their full save DC, which would most likely be not RAI.
Thus when using these items, you can use the ability linked to the class you chose for the feat to set the DC.
