IMHO "universalist wizard" is a weak character concept because there is no universal school in 5e and because it tells you nothing about the characters approach to magic.
"Hedge magic" or "scholarly magic" are both good (and different) concepts that capture the feel I think you're aiming for. Say we took "scholarly magic" as an example, here are some ideas...
Scholarly Tradition
Level 2 Ritual Savant: The gold and time you must spend to copy a ritual spell into your spell book is halved. In addition you can copy all rituals into your spell book regardless of class, though spells not of your class can only be cast as rituals.
Level 2 Cautious Reader: When reading writing or inscriptions, you never trigger any traps, curses, or spells contained within them unless you wish to do so. In addition when you read something you can cast
detect magic on just that writing without expending a spell slot and without the need for components.
Level 6 Potent Rituals: When you use a spell as a ritual, cast it as if you had expended a slot of one level higher than the spell's ritual. In addition, when casting spells which can be made permanent, halve the number of days you must consecutively cast the spell.
Level 10 Scholarly Insights: You can use your action to make connections between various pieces of lore you've gathered and make amazing deductions. When you do, choose one of the following benefits which lasts until you're incapacitated or you take a long or short rest. You can't use this feature again until you take a rest.
Greater Comprehension. You can read any language, or communicate in a very basic way in a language which shares a common alphabet as a language you already know.
Puzzle Solving. You gain advantage on all Intelligence checks to solve a puzzle, play a game of skill, or crack a code.
Rushed Ritual Casting. You can cast a ritual with just a few minutes of extra time (at most 5, possibly less if your DM allows). However, when you do so you cannot apply your Potent Ritualist benefit and there may be other complications at your DM's discretion.
Topical Knowledge. You can apply half your proficiency bonus to Intelligence checks to recall lore pertaining to skills you're not proficient in.
Level 14 True Grimoire: Your spell book is invested with power from years of your pouring over at pages, developing a familiar-like bond with you. This grants you the following benefits:
- You can direct your spell book to copy spells onto its pages without you needing to do so - a quill will magically transcribe a spell scroll into the book at your command with no need for your concentration or time (though you still must spend the cost for rare inks and other components needed).
- You can summon your spell book to your hand as long as you and it are on the same plane.
- You can get a sense for the immediate environment around your spell book by concentrating , though while doing so you no longer perceive your surroundings.
- If your spellbook is ever destroyed, you can speak a command word when in some grand library to cause a copy of your spellbook to mystically appear on the bookshelves. You will still need to pay 10 go for each spell level in the book to copy over everything with rare inks and make sure the formulae are right.
- Add two extra spells to your spell book, and every time you gain a level, add three spells to your spell book instead of two from now on.
