Firstly, the term "Turkish" doesn't belong here since no such thing as "Turkey" existed in that era. The Bulgarians were originally a Turkic tribe, and the word Bagatur is of Turkic origin, but not "Turkish".
Secondly, making it a cavalry archer is a bit bizarre. While the Bolghars who settled around the Volga likely did incorporate horse archers, the group that went to settle in the Balkans favored heavy-cavalry tactics, specifically a cavalry charge at the right time. It's written about extensively, as well as depicted in art from the period. See the below image (Bulgarians shown with swords).

Whoever was responsible seems not to have done a lot of research and may even be conflating the central Asian Bolghars with their descendants after the settling of the Volga, which is simply inaccurate since the Bulgarians who settled in Europe had long since abandoned many aspects of steppe culture. Horse archers were present in Bulgarian armies in moderate amounts, but these were primarily Pecheneg mercenaries and not part of the Bolghar ruling class, of which the Bagaturi were the highest military echelon.

There is a marked separation in time and place between the inspiration of this unit and the inspiration of the culture as a whole. It's like giving the Industrial Era Germans a Teutonic Knight as an emblematic unit.

Thank you for the explanation @FestinaLente, it's much appreciated.
I understand where the decision came from, but then the name of the unit "Bagaturi" seems misused. Since those were the Khans' most loyal, trusted and distinguished warriors, they would be "wholly armored with iron" as described by Eastern Roman historians during the pre-Christian era of the Bulgarian Empire. The unit should logically be renamed since the name is simply not reflective of the unit present in the game. 

When developing our own Medieval Bulgarians mod for a much earlier version of the game, we also ran into the issue of adding yet another charging heavy cavalry because we specifically wanted to add the Bagatur as a unit, so we opted for a defensive cavalry instead since the charge tactics were usually applied as a counter-offensive instead of at the start of a battle. Neither here nor there since there's no chance to change the unit, I'm just giving an example of our thought process that's somewhat more historical.

I  think than the arguments for a mounted ranged unit for bulgarians, and called Bagatur, make sense. The reasoning of the dev is fine.

The issues I see, from what highlighted kconstantine, but also when we make some research about medieval bulgarian warfare is than :

The artwork looks like depicting early and legendary bagatur among Volga Bulgars (which is a bit a stretch, and could just look like any Volga raider) , and not really the bagatur from first and second empire. And I am not that sure they were so few armored. Seems to be a niche take

Or it looks like the numerous mercenaries which got that role (mobile, light and ranged cavalry) in second empire (in that case Bagatur naming is not fitting, and they are kipchak-cumans mercenaries).

In term of representation we are in something really late classical, ala Hunnic Horde (in term of vibes). And even in Medieval it feels similar to Mongols which themselves used heavy hybrid cavalry (ranged & melee), but it's fine for Mongols trope to have the current unit.

What could offer Bulgarians different from Mongols, and still being considered as Bagaturi : An heavy ranged cavalry. The artwork go a bit more armored, the unit have few range and don't have the melee malus like archers. And reasonable stats. And u have an unit radically different from Huns and Mongols, which cover early Bulgar to second Empire.

The current design of the culture artwork for that era give more magyar vibes than bulgarians. But it's maybe just me.
Overall their design is still really nice imo, and I remember a bunch of poeple expecting some volga bulgars feature when Bulgarians were suggested into 4X games (like Civ), so shouldn't be noticeable for most of the community

Well a bunch of design cover the whole era, instead of a specific period. Like French only have as napoleonic their Emblematic Unit. Or Siamese and Persians have a really earlier quarter than their EU.

For the depictions of horse archery it probably need to be looked at the larger scale, but it's safe to assume than cultures which were using composite bows and able of horse archery, were equiped by a bow even when firstly an heavy cavalry. And can be observed on Mongols in medieval period, don't need to look at 16th century. Byzantine also had brigade of mounted archers but they were mercenaries. And you can take a look to the equipments of petchenegs or kipchak-cumans. Same mercenaries which were heavily used by bulgarian second empire.

Also there is this depiction of Bulgarians fighting the Rus' Invasion

So, would be bizarre than Bulgarians don't have access to mounted archers, I think it's expected and more unique than heavy cavalry which is avalaible for everyone.

But I was just agreeing than Bagatur naming was not totally fullfiled by the design of the unit (the artwork)

If bagatur are the "heroes" and elite, I expected something more ... armored, even if using a bow (I know these illustrations are not totally accurate but still)

To resume my nitpick, it would be like naming the current Mongols Horde unit : Khan, Keshik, or Mangudai