I honestly wish the Bard had been a half caster, expanding the bardic and combat abilities to make up the difference. To me they should start with medium armor proficiency and a few fighting style options (I'd go with TWF, Duelist, and Defense), being the warrior poets of legend.
Level 1 - Bardic Inspiration (1d6), Jack of All Trades
Level 2 - Fighting Style, Song of Rest (1d6), Spellcasting Lv1
Level 3 - Subclass Feature, Expertise (2 skills/tools), Font of Inspiration
Level 4 - Ability Score Improvement
Level 5 - Bardic Inspiration (1d8), Countercharm, Magical Secrets (2 Spells), Spellcasting Lv 2
Level 6 - Extra Attack, Expertise (3 Skills/Tools), Song of Rest (1d8)
Level 7 - Subclass Feature, Words of Valor
Level 8 - Ability Score Improvement
Level 9 - Spellcasting Lv 3, Magical Secrets (4 Spells)
Level 10 - Bardic Inspiration (1d10), Expertise (4 Skills/Tools)
Level 11 - New Ability, Song of Rest (1d10)
Level 12 - Ability Score Improvement
Level 13 - Magical Secrets (6 Spells), Spellcasting Lv 4
Level 14 - Subclass Feature, Bardic Inspiration (1d12)
Level 15 - Improved Words of Valor, Song of Rest (1d12)
Level 16 - Ability Score Improvement
Level 17 - Magical Secrets (8 Spells), Spellcasting Level 5
Level 18 - New Ability
Level 19 - Ability Score Improvement
Level 20 - Superior Inspiration
This is something more of what I'd like to see. The loss of primary spells does make them weaker, but they get quite a few of their abilities earlier than normal to make up for it. I moved Font of Inspiration back to level 3, because Bardic Inspiration would be the class's defining feature (which it already is). They pick up extra attack, but a level behind true warriors (I might actually consider moving it higher, but YMMV). They get more Magical Secrets, but with fewer and lower spells, this becomes more of a versatility option.
I've thought of a good ability that seems very "Bardic," which is similar to the Inspiring Leader feat (a common Feat for Bards IME). Basically you can spend 1 Minute to inspire a number of creatures equal to your Charisma modifier, then spend a Bardic Inspiration to give them that many Temporary HP. At level 15 you can double the number of targets and roll your Inspiration die twice for the number of THP. I honestly don't know how good this would be, as HP are really good, but you don't get that much. However, with short rest recovery of Inspiration, it can be done repeatedly. It isn't as good as the Feat (obviously), but it's shorter and has little/no opportunity cost.
There's 2 spots for new abilities that I can't think of. I considered at one point doubling the number of inspiration, but that feels it might be too powerful.