No they shouldn't unless they get unique spells.
The uniqueness come from their ability to mix spell lists.
All full casters should have a unique spell from level 1-9.
Over the years and editions bards have had many unique spells. WotC keeps giving them away to clerics and wizards. They don't need unique spells, however; they need a spell list that suits the flavor of the class.
Bards are only full casters because WOTC (and most of the D&D community) couldn't balance them as a 2/3 caster and refused to make unique bardmagic nor give it a flavor past mid levels.
That's an argument of presumption in that it's based on presuming bards should have been "2/3's casters" in the first place, which isn't true.
Bards were powerful casters in all editions. Not quite as powerful as other some other spellcasters but still on the powerful side of spell casting. 5e maintains that difference in other way. If we compare a bard to a cleric, druid, or wizard that's maintained through the spell preparation mechanic and subclass features. Arcane recovery alone places wizards above bards. If we compare a bard to a sorcerer that's maintained through metamagic and subclass features. Either innate sorcery or font of magic places sorcerers above bards.
If we're going to use an old spell progression table as the basis of what "should be" then also cut back clerics and druids to "2/3's casters", remove spells from warlocks and rely solely on invocations, and seriously gut spells from paladins and rangers to match.
3.5 bards casting mass suggestion over a dozen times per day was more magic than most 3.5 magic classes could hope to demonstrate. Combining the 3.5 bard songs and spells into one mechanic (spells) does make for a full caster, and we tried the half-caster before 5.14 -- it wasn't popular. ;-)
Bards were also powerful magicians in some mythology. It makes sense to have them as a powerful magician in a fantasy game to match. Anyone who wants to play them differently can take a few levels of fighter then a few levels of rogue then continue as a druid if they want. Or make and arcane trickster with the musician feat claiming to be a bard.
The only thing "should have been 2/3's caster does" is deny those other mythological bard archetypes. That's not necessary or desirable.
Monks in 2024 are crazy OP as everyone knows.
"More fun" and "OP" aren't the same thing. ;-)
And Bards should be half-casters. Never should be full-casters, never should get expertise by default.
Bards are more often depicted as dabblers in everything, not focused and dedicated enough to be full caster or have expertise by default.
See my comments above.
