1 thing to add: yes you can beat the Mongols with Scots.

Hvy Pike Militia are pretty arrow resistant. Noble Highland archers Ok, same range as HA and ok in melee (as long as not charged). Border Horse fast.

It's not cheap and it's not pretty but....

Put your Pikes in a horseshoe - i.e. make sure you have several units at flank and 1 to the rear. Pile your archers in the middle (with culverin if you have it). General safely in the middle.

The swarm come in: All your Pikes go red under fire. But they won't run and take fewer casualties than you first think under the hail. Meanwhile the NHA are returning fire. If you can turn your artillery on the mongol foot archers all the better.

Now here's the good bit: As long as you can hold firm the Mongols will charge you. Their big nasty general: Bam!! Shish kebab. Their HA, their lancers - all will just disintegrate against your Pikes. Once they've been sliced to ribbons, now you can send your cav against the Foot archers.

You do take losses but hvy pike militia are dirt cheap and available in Large (not even huge) cities.

Obviously if you're at a river or bridge then you can win with almost no losses at all. I had an entire mongol elite army charge into oblivion and didn't even rotate the two pike units holding the bridge (in a V to allow the mongols in a little, making the killing ground even sweeter). I lost almost nobody.