Given that 5e yanked everyone's AC way down from earlier editions, I don't know that light armor is the biggest limitation, but yes, it's probably a bad idea for a bard to be in greatclub reach, anyway.
Sorcerers and Wizards: No Armor training.
Bards and Warlocks: Light Amor training.
Druids: Light Armor and Shield training.
Clerics: Medium Armor and Shield training.
Light armor isn't much of an AC boost and Warlocks can be squishy too if they aren't careful. +1 Armor is Rare, +2 Armor is Very Rare, and +3 Armor is Legendary so randomly finding armor of the right category with a bonus is a small chance. Even the "Starting at Higher Levels" rules wouldn't allow for +1 Armor until 11th level.
Useful Magic Armor tends to be deliberately awarded by the DM or crafted by players if they have the resources and downtime.
Sorcerers and Wizards can get by with Mage Armor, Shield, and Blade Ward as options. Sorcerer spell preparation limits them at 1st level but they have more cantrips and not much use for concentration yet.
Warlocks can replace their armor with Armor of Shadows at the cost of an invocation but don't have Shield and might not have room for Blade Ward, plus Hex is likely to compete for concentration early on. Armor of Agathys can help, and the the Fiendish Vigor invocation can go a long way early to help with squishiness because it's 12 temp hp at will.
Those defensive options come at the cost of offensive options or utility, however.
Druids have a choice of Magician or Warden. Magician gives the WIS bonus to Arcana or Nature checks, and a bonus cantrip. Warden gives Medium Armor training and Martial Weapon Proficiency. Just a shield is an improvement, but all Druids can access healing spells if they prepare them on any give day and Wild Shape grants temporary hp. The choice for Medium Armor training can help as well but it depends on how the player expects to play, particularly in respect to Wild Shape.
Clerics have a choice of Thaumaturge or Protector. Thaumaturge gives the WIS bonus to Arcana or Religion checks, and a bonus cantrip. Protector gives Heavy Armor training and Martial Weapon Proficiency. Like Druids, Clerics gain a solid benefit from just the shield and can access healing spells if they prepare them on any given day. Clerics start with better armor options than other spellcasters, can improve AC with Shield of Faith, and can use Channel Divinity for additional healing.
Bards don't have the the inherent amor benefits of a Cleric or Druid, or inherently have defensive spells like Mage Armor or Shield like a Sorcerer or Wizard, and they cannot cast False Life on themselves with full temp hp at will through an invocation like Warlocks. A lot of their base abilities help others instead of themselves as well.
Bards can make choices that help, but they're usually playing catch up after the other spellcasters in this regard.
A Valor Bard in half-plate with a shield using Blade Ward because they didn't take other concentration spells who adds the Shield spell at 10th level is exactly squishy at that point. At that point the PC is combining decent armor benefits with spell support and makes for a decent support character up at the front line.
A Dance Bard wearing no armor likely has 16 AC from Unarmored Defense at 3rd level and 18 AC by 8th level, and can also rely on Blade Ward instead of using a lot of concentration spells and pick up Shield at 10th level, plus adds mobility from Inspiring Movement to avoid opponents.
A Glamour Bard doesn't get the AC benefits and Mantle of Majesty requires concentration. They tend to stay squishier. My current Bard is Glamour and squishy, but worth it (IMO) because of the focus on Enchantments and Illusions in the Glamour College. He's kind of squishy.
A Lore Bard might use Cutting Words to deflect attacks but that will eat through Bardic Inspiration fast. They also tend to remain squishy. But they can add better spell support sooner.
tldr; Every PC has to deal with defense somehow. "Squishy" still happens in some cases. ;-)
