In fairness, that misconception doesn’t just spring from imagination. Prop weapons generally are heavy, because they need to be thick to have blunt edges.
$10 halloween costume prop? Hollow plastic, super light.
Training weapon for self defence against knife attacks? Solid plastic, several times the thickness of a real knife, probably bright blue.
Olympic fencing weapons, which simulate duelling? Pretty light, you only need a light touch to score a point and bending is desirable.
Live Action Role Play where they want to whack each other, but not hard? Anime cosplay? They mostly go for foam-covered plastic which is kinda light, but they're often thick and also they often simulate unrealistically large weapons, so they can end up kinda heavy.
Ornamental/replica things that are just for show? Could go either way. You could make an extra-light sword out of aluminium, or an extra-heavy ornate dagger out of pewter. Less steel means a lower material and shipping cost - but a bit more heft that feels realistic to buyers might get you better product reviews.
Apparently some HEMA/renaissance fair folk wear full suits of metal armour, metal longswords that are blunt-but-authentically-weighted, then try to whack each other pretty hard? Looks like concussion city to me.
Forget Under Armour. Let’s go for Over Armor.
I have a full hauberk which goes to my knees with long sleeves. It weighs 6-ish kilo. Add a kilo for the coif. A decently made late medieval full plate armour is maybe 15-ish kilo complete with helmet. The heaviest armours are probably late 14th century armours. They are essentially an almost full plate over a (partial or full) maille hauberk. Some are maybe 25 kilo. That's still a lot less than the 40 or so kilo that modern soldiers lug around.
https://takknife.com/collections/keen-edge-aluminum-training...
For example, here's a practice sword that has 1/16th in (1.6mm!) thick edges: https://www.reliks.com/functional-european-swords/practical-...
With the degree of difference in edge thickness, you can probably see how the cross section of a sword (and therefore its weight) will vary substantially, especially if you keep the maximum thickness about the same.
It's absolutely possible to craft these types of swords that are both safe and well approximate the real shapes and weights and handling of a "real" sword. But it's also easy to just... go for look and safety to try to save money.