One problem. Thrust speeds dictate which ships can engage and kill which other ships. For example, a gunboat on its own can't reasonably kill a lone fighter, because the fighter can simply thrust away. Sure, the gunboat can cruise up, but during the charge sequence the fighter can charge its very own cruise, and the engagement will be over.
The same would happen with heavy- and light fighters. They'd simply be able to run from everything except their own class of ships. It'd be absolutely trivial to shieldrun away from something you out-thrust. Let's be honest, it can be hard enough to catch up to a shieldrunning fighter (provided they know what they're doing and don't dodge like mad) that's just out of effective weapons range as it is, with the exact same thrust speed.
The current system for heavy fighter balance is fine, anyways. It's just not very consistent. There are ships like the Bayonet, which is an absolute beast in a duel if flown correctly, but there are ships like the Executioner - an Avenger-sized ship with a paper-thin hull and agility that doesn't quite match that of the Bayonet, which also happens to have a preferable profile.
Isn't that the point of the "scout" role of the lights and even somewhat of the heavy's? Shouldn't a light have the ability to harass the heck out of it's larger counterparts? Reductions in damage output and armor are rewarded with both speed and maneuverability in capital ships. Why not snubs?
Shield runners reduce their own effectiveness by not engaging anyways. If you used that speed only to dodge in an LF, you're a fly not worthy of the time it would take to swat. Effective usage of that speed would be in full retreat, possibly kiting, and pursuit of a heavier fighter. It would natively provide an benefit to avoidance.
The biggest advantage of taking a lighter fighter should be the ability to "choose" your own engagements, and avoid the ones that don't play in your favor.