Militia vs Trained

[Kraxis]

Yes we agree on the first part.

But I happen to sit with Livy's account right here, and it seems I have been mising a little line that seems confusing at best.

The cavalry and light infantry who had been in action he stationed on his right; the caetrati and the men of the phalanx were ordered to lay aside their spears, the length of which only embarrassed them,

So he does account an order to let the spears drop and to double the lines. And the losses amounted to 13000, but that is killed (which presumably includes the wounded who were likely put out of their miseries) and prisoners, and they amounted to 5000 in all. So 8000 were killed.

However, what Polybius has to say might shed some light on the matter.

Upon this being done, the enemy being now close upon them, orders were sent out to the men of the phalanx to lower their spears and charge,

Lower their spears and charge... Makes sense eh? Meanwhile it can also be misunderstood when you read one language as complex as Greek and write in another (Latin). So it seems Livy simply misunderstood Polybius, for he clearly states later that he used Polybius as his source because he was by far the best when it came to Greek matters.
This one would also seem odd if they didn't use the pikes.

the nature of their [the Macedonians*] arms also giving them a decided advantage on the present occasion

*- Insert by me.
A smaller shield and smaller weapon would not confer any sort of advantage in such a fight. A small shield would be better in an open more personal fight, but where total protection was needed and movement was impossible, such a shield was less than the scutum in effect. And while the enchiridion was a large dagger or small short sword, it was a slashing weapon, not a stabbing weapon (basically it looked like a small kopis/falcata). That would not be terribly great in such a fight anyway.

I think it is clear that Livy have made one of his rather numerous errors (he was a good authority on knowing sources, but apparently a bad translator).

About the left and it being them who raised their pikes.

Most of the Romans followed up these fugitives and continued to put them to the sword

General chase as we know them. Merciless, if not terribly effective. But the fact remains these troops were broken (literally as there was no semblance of formation left), but that it is also likely that they were the ones who made it away from the battle (they fled before contact).

Meanwhile after the right broke, it seems Flaminius stumbled upon a unit of the phalanx who had been somewhere between the left column and the right line. They were formed and on the summit (a position the left only barely reached before being on flying).

When he noticed that the Romans in pursuit of his left wing had already reached the summits, he decided to fly, collecting hastily as many Thracians and Macedonians as he could. Flamininus, pursuing the fugitives and finding when he reached the crest of the ridge that the ranks of the Macedonian left were just attaining the summits, at first halted. 10 The enemy were now holding up their spears, as is the Macedonian custom when they either surrender or go over to the enemy, and on learning the significance of this he kept back his men, thinking to spare the beaten force. But while he was still making up his mind some of the Romans who had advanced further fell on them from above and began to cut them down. Most of them perished, a very few escaping after throwing away their shields.

Take not that the quotes are in chronological order, so the left has already been routed a good time ago, and is being actively chased by the Roman right. Apparently one taxis survived, distanced from the others, or else it was the remains of the right, or possibly even a sort of reserve, many possibilities. It also seems that some of the Roman right turns back and attacks them from behind (how they can attack from above when the unit was at the summit isnot explained) and kills them (clearly they didn't become prisoners). In any case a decidedly odd incident.

Polybius gives Livy his numbers on casualties, 8000 dead and at least 5000 prisoners.

About the short spear. A short spear would be around 5-6 feet, easily turned and used, and spears are fast weapons with good properties of penetration, hence a short spearman could inflict as much damage as swordsman.
A lot has been said about celtic swords, but they had by this time abandoned the blunted sword, and used a sword that did both cut and stab. When such a swordsman got to the phalanx front he wouldn't need to get in and play Roman legionary, he would then be able to hold his distance. At that point on a single point would oppose him, wielded by a guy who could hardly see what was going on, and at this point the other 5-rankers could not turn their point to help halt another file because their own file were in the way of the move. The swordsman would thus be able to stand just outside and chop up the phalangites with wide cuts they could not deflect (unless they dropped the pike naturally) and stabs they could not respond to. But that didn't happen because the Celts failed to get people inside the pikes.

About shields (I'm beginning to sound like one of those ancient historians with those headers ).
The republican shield was roughly shaped as the shield the Hastati, Principes and Triarii have in RTW. Super-eliptic (more or less rectangular with round corners) and curved. And instead of strengtheners (like the 'L's I mentioned before) it had a central vertical spine encompassing the boss. This was apparently for strength as well as decorative functions. The shield with a spine would not be bent back too far and would be more resistant to powerful strikes, but would likely suffer more structural damage in a fight.
And Celts, unless they were skirmishers, tended to use large oval shields. A shield is relative cheap, and can easily be made and replaced compared to arms and armour. Shield and spear was a requirement for most. Armour was a benefit of station and rank.