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General idea
If the intention is to try and build a more realistic representation of trade companies in the game, I'd recommend giving this a read: https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/colonization-and-companies
It's a summarized history of trade companies and some of their mechanisms, and it highlights the various differentiating aspects and evolution of various types of trade companies over time.
Whether and which parts could be modelled in game is a different conversation. Personally, I'd go with what "feels" good to play, rather than historical representation.
So what role should a trade company play?
Given that the regular colonial subjects already give a fraction of trade capacity and advantage to the overlord, perhaps a trade company could be a more specialized type of subject, focused on moving goods to the overlord's market?
In my mind, trade companies would be a better representation of the "burgher importing" and "burgher exporting" parts of your trade. If not necessarily replacing that entirely (due to game balance), enhancing it, such as your burghers being able to import and export rarer goods, for cheaper (than you can), from further away. Something you don't have direct control over, but which serves the market needs and ultimately benefits your nation.
The above would require that mechanism to be exposed for modding, however, and so far, I haven't seen any way to control this aspect of trade in the game files, so I don't know if it's feasible to tap into it right now.
Trade and income
I'm thinking here that a trade company wouldn't have much of a tax base, generally. With some exceptions, I don't think they necessarily would own most of the valuable province RGOs, but instead try to siphon off the goods and send them to the home market, as well as import from the home market and sell to their own.
For this, they'd obviously need a decent trade advantage and capacity. A baseline should be given by their own buildings, but I'd suggest also having a bonus of these based on the overlord's own. After all, a more influential nation could extend that influence overseas and their companies would be more competitive compared to a smaller or less trade-focused nation.
In terms of how they'd make money, my thoughts would be taking a percentage of all trade profits made by the company (the resulting sum being boosted by some buildings and special modifiers to avoid making trades unprofitable while keeping a decent income).
It would be a careful balance of numbers, so maybe your burghers could also subsidize them a bit.
Given that these subjects are more of an extension of your burgher class, and not really directly controlled by the crown (at least if we're going the semi-realistic route), I'd say they should be able to maintain a small regular force, and maybe being able to hire mercenaries at a discount if necessary.
Given the above, I'm actually not convinced that them being declared on, or declaring war, on a local nation (in the same subcontinent as them, as opposed to your own) should drag the overlord into that war. I'm of the opinion that local conflicts should stay local, but that might create some serious survivability issues. Perhaps they should have a diplomatic bonus to reduce the chance of conflicts?
Effect on the host country
As much as EU5 loves its modifiers, I see things a bit differently here. A trade company's impact on its overlord wouldn't be direct, through modifiers (maybe just a bit) but mostly through the market influence and the goods it's transporting. If it could fill in market gaps where you, as a player, can't easily, it would be a great incentive to create some, even if they aren't as directly profitable or give as many direct bonuses as other subject types.
Think about importing valuable ivory, amber, spices, porcelain, etc. Stuff that you'd struggle to do without expanding across continents and directly controlling provinces, building markets, etc.
In general, no real tax base, making trade companies the best entities at conducting trade, and providing part of their profits to the crown. One concern I do have is the trade companies competing with the overlord in a detrimental way, for example, by outcompeting your trading posts for valuable trades. I have a few ideas for this.
1. Prevent the overlord from building trade outposts in the same locations as the company. And the trade company itself would make up for this by having a much greater trade capacity and advantage, ultimately proving more profitable. i.e., 50% of the profit from 100 capacity of more advantaged trade exceeding the 100% profit from 30 capacity of trade.
2. Leave overlord trade outposts alone and have the trade companies provide a greater trade capacity to overlord than themselves (but not capacity). This would more simulate the crown forcing the trade company to serve their needs first. So that the overlord would have far less trade capacity in the region than the company, but the little capacity they have would take the advantageous trades first.
3. Leave the overlord trade outposts alone and don't provide any additional trade bonuses to the crown. Essentially, just let the trade company cannibalize the profits of the overlord with their greater trade advantage, but still allow the overlord to have their own trade buildings around anyway. But it would still be made in a way that you will have greater value from the trade company than without.
In general, I prefer 1; it also seems to make more sense given this being a crown-established trade monopoly. But there is a potential downside if you are unable to manually trade goods to fix a missing need, for example, if your ships won't repair because you are lacking equipment. I think it would probably be fine, as with enough capacity, the trade companies will probably fill any needs like that, but there is always a chance.
This actually might be something that I could allow the overlord nation to dynamically set, such as declaring a company a trade monopoly, and therefore transferring all crown trade advantage and capacity to them. Then, having ways to achieve each of the other listed scenarios, but I'm not certain.
Additionally, while reading through the page you linked, I wasn't aware of some of the different kinds of trade companies around at the time, such as "the English Russia (or Muscovy) Company (1553) and the Levant (or Turkey) Company (1581)" regulated companies. These didn't field any private armies like the later joint-stock companies, and acted much more like merchant guilds, rather than a united entity, but reading up, they still had a substantial effect on and benefit to the crown's trade. So it could be interesting to add a separate earlier unlock company type for it. If so, this would definitely have to come later, but I wanted to bring it up while I had it on my mind.