As something of an amateur archer myself I've thought about this.
I can feel you there. I've spent way too much time thinking about game stats for bows. I consider myself an archaic archery enthusiast, and pretty much always choose to play archers in D&D or any other fantasy game.
In truth, if one wanted to simulate the fact that stronger bows would require a stronger person to use effectively, bows would come in several categories based on their draw weight. Game stats for this would most likely come in the form of required Strength Score (or even just required Strength Modifier). Each category of bow could then have increasing damage die (or dice). Of course as the thickness and draw weight of the bow goes up, so too would the cost of the bow.
I'm actually perfectly fine with Dexterity being used for both the attack roll and the damage roll with the caveat that Strength would be used to determine the range that you can fire the bow. Something like Strength score x5 for short range and x10 for long range or something along those lines. Keep it simple.
Where the real complexity comes in is based on the ways that ancient cultures modified their bows, the two most important being Compositing (adding bone or hide or other material to the back of the bow to add strength) and Recurving (bending the bow arms over the back). According to the majority of the research you can find both procedures are generally believed to have allowed for greater damage and distance from your bow allowing for smaller bows to do more damage, thus allowing many horse-riding (or even raiding) cultures to make use of smaller yet still powerful bows.
Of course how to turn this into a simple and easy to calculate game mechanic is the real trick as none of us wants to turn 5e into GURPS. (Well I would hope none of us would...nothing against GURPS.) I would suggest either simply adding a bonus to range like +30 ft or something arbitrary and adding +1 or +2 to damage.
Putting that all together, a quick and dirty list off the top of my head looks like:
STR
Requirement......Damage......Special
11 or less............1d6............No disadvantage while mounted
12 - 13................1d8............No disadvantage while mounted
14 - 15................2d4............Disadvantage while mounted
16 - 17................1d10..........Disadvantage while mounted
18+....................2d6............Disadvantage while mounted
Range of all bows: Strength Score x5 / x10
Recurving: Add +1 to the multiplier for both Short and Long ranges of the bow (example x5/x10 becomes x6/x11). Add +1 damage. Recurving must be done when the bow is made and cannot be done afterwards. Add 50% to the cost of the base bow.
Compositing: Add +1 to the multiplier for both Short and Long ranges of the bow (example x5/x10 becomes x6/x11). Add +1 damage. Compositing can be done after the bow is made. Costs 50% as the cost of the base bow.
This of course may be way more than anyone wants to deal with especially given the simplicity of 5e, but I think it could add a bit more realism without strictly going for true simulation.