RAW any damage is a roll

As you have noted, a critical permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that all damage is defined as a die or dice. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is always a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and then apply the total damage to your target'. For an unarmed strike that does a base of 1 damage, there is no RAW way to apply this amount as damage unless the value of 1 represents a roll.

For damage that is specified as a flat amount x, the damage die must be of the form xd1. An unarmed strike (or blowgun needle, or sprite longsword) which does 1 point before the Strength modifier is thus a 1d1, whose only outcome is 1 damage. On a critical the damage dice would be doubled to 2d1, or 2.

At least, this is what we can conclude from the language in the PHB. Is this what is intended? Most likely not, and see below. But it is what is written.

RAI flat values are not rolls

As Jeremy Crawford tweeted,

An unarmed strike deals 1 + Str. modifier damage, even on a critical hit.

The intention is that an unrolled damage amount is not increased on a critical.

Roll20 treats flat damage values as rolls

If you use roll20, you may notice that a flat value is treated as a damage roll. For example, if your character is equipped with a blowgun, the needle will do 1 point of damage (plus your Dex mod if you are proficient). But on a critical hit, it will display the damage as 1 + 1.

That the calculation of damage for a critical hit is applied in this manner in roll20 says nothing about RAW or RAI for the rule, but it is worth noting in case this application differs from the ruling at your table.

The general rule is that damage is a roll

As you have noted, a critical permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that damage is defined as a die or dice. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is always a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and then apply the total damage to your target'.

Unarmed strikes are an exception

The PHB section on melee attacks says:

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.

The specific rule about unarmed strikes overrides the general rule that damage is rolled. Because there are no damage dice to roll for an unarmed strike, a critical hit has no dice to double. A critical hit on an unexceptional unarmed attack does no extra damage.

RAW any damage is a roll

As you have noted, a critical permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that all damage is defined as a die or dice. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Any damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is always a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and apply the damage to your target'.

For damage that is specified as a flat amount, such as an unarmed strike, blowgun needle, or sprite longsword, this damage die must be a 1d1, whose only outcome is 1 damage - but which on a critical would be 2d1, or 2.

At least, this is what we can conclude from the language in the PHB. Is this what is intended? Most likely not, and see below. But it is what is written.

RAI flat values are not rolls

As Jeremy Crawford tweeted,

An unarmed strike deals 1 + Str. modifier damage, even on a critical hit.

The intention is that an unrolled damage amount is not increased on a critical.

Roll20 treats flat damage values as rolls

If you use roll20, you may notice that a flat value is treated as a damage roll. For example, if your character is equipped with a blowgun, the needle will do 1 point of damage (plus your Dex mod if you are proficient). But on a critical hit, it will display the damage as 1 + 1.

RAW any damage is a roll

As you have noted, a critical permits an extra roll of damage dice, but does not change any modifications to the roll. That might make it seem like 'flat damage' is not augmented by a critical, but note that all damage is defined as a die or dice. From the PHB on Damage Rolls (emphasis mine):

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.

Damage requires a roll of a damage die, or dice, by definition. Modifiers are qualified as 'any', indicating that they may or may not be present, but no such qualification is made for damage dice. Damage is always a roll; we are not told 'You take any flat damage value, roll and add any damage die or dice, add any modifiers, and then apply the total damage to your target'. For an unarmed strike that does a base of 1 damage, there is no RAW way to apply this amount as damage unless the value of 1 represents a roll.

For damage that is specified as a flat amount x, the damage die must be of the form xd1. An unarmed strike (or blowgun needle, or sprite longsword) which does 1 point before the Strength modifier is thus a 1d1, whose only outcome is 1 damage. On a critical the damage dice would be doubled to 2d1, or 2.

At least, this is what we can conclude from the language in the PHB. Is this what is intended? Most likely not, and see below. But it is what is written.

RAI flat values are not rolls

As Jeremy Crawford tweeted,

An unarmed strike deals 1 + Str. modifier damage, even on a critical hit.

The intention is that an unrolled damage amount is not increased on a critical.

Roll20 treats flat damage values as rolls

If you use roll20, you may notice that a flat value is treated as a damage roll. For example, if your character is equipped with a blowgun, the needle will do 1 point of damage (plus your Dex mod if you are proficient). But on a critical hit, it will display the damage as 1 + 1.

That the calculation of damage for a critical hit is applied in this manner in roll20 says nothing about RAW or RAI for the rule, but it is worth noting in case this application differs from the ruling at your table.