6000 gp can run out quickly.
There are a lot of ways for PCs to spend down 6000 gp in the course of their adventures:
- Spellcasting: Lots of spells have expensive material components. Raise Dead costs 500gp, Hallow, Heroes' Feast, and Forbiddance (among others) costs 1000gp, and in a few levels, your players will have access to increasingly expensive spells like Sequester, which costs 5000gp. Hell, Planar Ally can cost 100gp a minute! The Downtime UA shows that making scrolls is quite expensive as well. These costs can easily eat through that starting gold.
- Equipment: While most adventuring gear is pretty cheap, some of it is not. Plate armor is 1500 gp, half plate is 750gp (PHB 145), and the infamous spyglass is 1000gp (PHB 150).
- Other goods: PHB 157 has a list of other various things your PCs might be interested in having. They could easily spend a few hundred gold on mounts, and most watercraft are more than 10,000 gp.
- Real Estate: DMG 128 shows that building even a small building can cost many thousands of gold. An outpost or fort costs 15,000 gp, for example. If your PCs want to have some sort of home base, it could be a huge gold sink. Moreover, such buildings also have persistent maintenance costs (DMG 127) that accumulate daily and can really add up.
You can also introduce custom costs. For example, you could give extra languages or proficiencies for gold. A "skilled" hireling is 2gp a day, but that might only be a 1st level fighter--hiring a spellcaster or a high-level hireling might cost a lot more. After all, at that point, your PCs are basically quest givers.
Based on costs, this sum might not "feel" exorbitant
It seems like you're concerned that the PCs will perceive themselves to have a massive fortune. However, based on the costs given above, they might not actually feel that way. Sure, that much money is an incredible amount for a commoner to have, but it's very far from an infinite supply. Even if they only end up spending a few hundred of that gold, they probably won't feel flush with cash.
For example, in my own campaign, some of my level 10 PCs are walking around with 15,000 gp, but they are unconcerned about it: it basically means that they don't worry about everyday costs. Of course, this is down to how your players feel and the style of your game, but I think it won't be much of an issue in a typical heroic fantasy game.