I would expect it to be a case of both, archery and/or men at arms companies would have been retained since the beginning of the gunpowder era and if still effective would have retained to Napoleonic era where I focus on the British experience as it is to be the British and KGL represented in the OP.
Thank you for the clarification.
The reason why they weren't used was because they would be rather useless, especially on the late 19th century. You can't use a bow while its raining; you have to create a industry to keep the archers arrow reserves high; they might fire 12 arrows a minute indicates that if they had the standard number of shot a musketeer had at the time, 30-40 shots, they would use their ammunition in 3 minutes; a musket had more penetrative power; and psychologically a musket firing is much more terrifying that you shooting a bow. Now that all of this and replace the musket for the bolt-action rifles that the armies used on the late 19th century.
Men-at-arms gave other reasons.
First off you can stop thinking of them as heavy cavalry. Sure they are well trained to one or one combat but the government would prefer to have their own heavy cavalry regiments that would be specifically trained to serve the tactics the army is using.
There is no need for heavy infantry on the age of gunshot. It would be impossible for most of them to have full armour, and the government ain't going to pay for it, so most would only have a breastplate, if they had the means, if not mail and letter, and a heavy musket can punch throw the breastplate while an arrow can't. Also see from the point of view of the poor bastard armed with a close quarters weapon when most of the killing is being done at long range; the guy will throw away his weapons take a musket and shot from some dead and fire back. Plus if you ordered heavy infantry to attack head on against a line of infantry who long do you think it will take until they break? No one is suicidal and advancing with just a plate and a sword/axe/[insert weapon] while the other side if firing lead balls against you those men would just run away.
Now no doubt you are thinking but what about on a defensive role? Surely heavy infantry would be useful defending.
Well sure but they would only be useful on that role, and the armies already had heavy cavalry to close gaps on the line, and to gain time to allow the reserves to reach the gap, so having 1 unit that is only useful on one scenario (men-at-arms) while the other (heavy cavalry) can do the same as the others and it can also be used to fight back against other soldier types.
Long story short, horse artillery takes a long time to train, horses for artillery service take a long time to train (that is true even in batteries where the gunners typically walk in battle) and they need riding horses besides taking advantage of those is a skill that needs teaching to most gunnery recruits. Light infantry and riflemen in particular needed more training than line infantry.
Long story short they were used, despite their longer training times, because they were useful. No 19th century general would consider committing an attack without having light infantry to test the other side; rifles were specific units on the early 19th century, but they constantly proved their worth during the Napoleonic Wars (during Badajoz they fended back several French raids, in San Marcial a company of 70 held a full French division for some hours under rainy weather, etc...), they were extremely popular among the army and they fulfilled they role of acting as skirmishers, sharpshooters and scouts perfectly.
The light infantry was also very useful. Look at Valmy where the Prussian line infantry failed to break the French Tirailleur.
They were part of the armies because they were useful and in some tactics vital. Horse artillery was needed to protect cavalry and their fast movement was vital for any tactic/strategy that relied on speed. They only became obsolete when the power of the infantry and of the foot artillery made cavalry obsolete, with the rise of tanks and self-propelled artillery.
Light infantry was a vital part of any army and it was the line infantry that died out, as line tactics became obsolete all infantry in practice became light infantry.
Riflemen were the specialized infantry of their time, the same way commandos are the special infantry of our age. Both are expensive to train and it takes a huge amount of time but for some reason but exist at different points in history because they provided more advantages than problems.
The man at arms in English usage might well be a mercenary but there is no requirement he be either a knight or nobleman. To pluck examples from my own family tree he might be a miller or son of a miller, not exactly your poor folk but yeomanry, craftsmen and merchant backgrounds, some men at arms in English service were also at times longbow archers. Contrary to the pike being an easy weapon to master it was in fact requiring of considerable strength in an age where such strength could not be taken for granted, indeed gentleman rankers chose to serve as pikemen because they were considered an elite compared to the mass of musketeers who from the English Civil Wars (Scottish and Irish flavours also available) outnumbered them two to one.
First the longbow is a weapon that takes long periods of time to master. It will take years to get a bowmen and the longbow requires lots of strength to use correctly. I never used a pike but I have experience with longbows of several heights and I can tell you that you need much strength to keep heavy bows still while aiming.
Also I don't know about English pikemen but most Spanish armies preferred to use mercenaries and/or voluteers to act as pikemen, with the second/third sons of the nobility and ruined nobles being known for serving in the ranks, usually to escape from their debts, and they served most of the times on the "piques armados" armed pikemen, the heavy pikemen. Not because they were the elite but because they usually brought heavy armour with them when they enlisted.
By the Napoleonic era the British liked to take six months to train a new recruit, this was much longer than most armies on the continent but still proved fairly robust in the face of massacre. However as noted there were specialist troop types who received even more training. Further but the English were not above for example apprenticing boys to military or near military service or naval service in particular where ship's officers would first go to sea as captain's servants or midshipmen to learn the profession. Besides how long does it take to train a man at arms compared to a nineteenth century cavalryman? Really, if you want them you can train them. Again for archers it is very rare for the enthusiasts who today we rely on to 'confirm' the feats of archery described in historical texts to have actually trained from boyhood to the role. Selecting from the population for stronger, more fit (as in aptitudes in this instance) individuals for specialised tasks is actually a quality that governments (not just the English government and the Scots) improved at over the ages.
Further we have the question as to why the English government did relinquish its laws on archery and no longer see the necessity of compelling its gentry to train as men as arms?
The British could allow themselves to spend so much time training recruits because they never suffered a crushing defeat during the Peninsular War. Had that occurred they would do the same the other powers did, spend 2-3 months training a 3 battalions and after that they were ready.
Ok answer me on this question - Why do you think that the common men-at-arms would be better than a cavalrymen? The heavy cavalry is an evolution of the men-at-arms. The men-at-arms was trained to fight as an individual while the 19th century cavalrymen was trained to serve in a regiment, to fight as a unit and to fulfill roles that the men-at-arms didn't, like scouting.
Again man, the archer was abandoned because a longbow can't pierce throw good plate, the musket is easier to use and train and an archer would demand more ammunition than a musketeer.
If you send archers to combat in the Napoleonic Wars what do you think would happen? They would require much more protection from cavalry, than the line infantry, as the later had bayonets to protect against cavalry, they would require much more ammunition and if it rained they would be useless because the string would become wet, sure the same problem occurs with the gunpowder of the musketeers but the musket could, and would, be improved to remove this problem.
In the end the musket proved to be easier to use. You don't need specific laws to create a manpower reserve, its easy to mass train men with it, they were more deadly and had a far more brutal psychological effect.
Further we have the question as to why the English government did relinquish its laws on archery and no longer see the necessity of compelling its gentry to train as men as arms?
Because the people weren't training? I have no idea so it could be by any reason.