Well the original (and more traditional and historically accurate) bard was a "fighter/DRUID/thief + music thing." Not "mage."
So for starters, there, I would make a change.
You're definitely on the right track with cutting them back from a "primary caster" to a mixed bag/built-in multiclassing. For 5e purposes, I think 5e really dropped the ball making Paladins cleric/divine magic half-casters, Rangers druid/nature magic half-casters, and NOT making Bards the mage/arcane magic half-casters (yes, I know what I said above, but D&D has been having bards use arcane magic, exclusively for a long time. So that's what people expect...and I'm going to give them some of that. Bear with me.)
For 5e, my "iconic" bard would look something like this...
--Initial Armor + Weapons as written.
--Skills: Must have Perform or Persuasion, at least 1 Lore, and any other 2 skills.
At 1st level: -They gain Cantrips at 1st level, but otherwise use a Half-Caster Progression: Choose from Druid list, plus any Enchantment or Illusion spell/magic type from any class' list.
--Bardic Inspiration, as normal.
At 2nd: Fighting Style, Spellcasting kicks in on the paladin spell progression (rangers have spells known, whereas the paladin can just choose from their lists...I believe the latter is more flavorful for someone of a bard's diversity. If they have a situation, they can come up with a song or chant or incantation to fit the situation vs. being locked into the 2-3 songs per level they know), and Jack-of-all-Trades.
At 3rd: Bardic College (choosing the Valor college gets you Medium armors and more martial weapons), Expertise
...huh...looks like, basically, from there on out you can just follow the bardic features progression (paladin spell progression table) as normal.
In my own homebrew, Bards are placed in the "Mystic [nee "Priest"]" class category. This along with other things, means:
--their HD is set to d8
--they gain access to Light and Medium armors immediately. Unlike other Mystic classes, I remove Shields as an option.
--have a limited array of weapons. In the bard's case, this includes all Simple, all Ranged, and light single-handed weapons (which immediately allows for short swords, scimitars or rapiers)...giving them, essentially, the same as the Rogue category of classes.
...and perhaps most notably:
--their primary class shtick is based around "channeling" with spell-use (access to a full spell list and slot independent progression) not kicking in until 3rd level. In the case of Bards, this falls under the same umbrella as druids as "Channel Nature."
So, at first level, the Bard gains the following channeling abilities:
--Sense Balance (a general neutral-aligned version of "divine sense" that reveals extraplanar entities or areas/person/objects of extreme alignment -LG or CE)
--Fascinate -ye olde area effect charm to neutralize opponents/distract attentions
--Inspire Competence (which essentially works similar to Inspiration dice, granting roll bonuses to others skill and save rolls).
Also at first level, the Bard can use a couple of cantrips per day (my homebrew doesn't have at-will cantrips, until 12th level for bards) and a "Jack of all trades" type bonus to skill rolls.
The spell list, when it kicks in (at 3rd), includes nature magic (which includes some minor healing), illusions, and enchantments. The Mystic class spell progression, basically, grants slots/progression (roughly) a level behind where mages and other arcane casters receive them. So, not exactly "half caster" in the 5e sense, but in my homebrew system seems to work out.
Additional channeling features as the class levels up include things such as the Countercharm, Inspiring Courage (bonuses to attacks and damage), Inspiring Greatness (better bonuses to everything).
Additional class features allow the addition of a martial weapon or two, and "Lore Expertise/Mastery" that allows a bonus to knowledge rolls for the flavor of ye olde 'legend lore" ability to identify items and/or recall stories/info about places and people of the 1 & 2e bards.
That about covers it.
Have fun with it.