Best to sell furnished or unfurnished?

I would be wary of spending money on furniture someone else may not like. Usually people make it their own home anyway. I decorated a house once prior to sale, new owner moved in and discarded it redecorating to their own taste. I don't see furniture adding value to a house sale

If your property doesn't look like a lovely place to move in and use exactly as is then better to empty it as it will look bigger and present a blank canvas for the viewer's imagination to paint their ideas in their mind.

In your own words, the 'rather worn furniture' will not leave a good impression.

I'm a viewing agent and a home stager and it very much depends on location, price point and demographic of the potential purchaser. If the property is in an area more likely to be purchased as a B2L remove the furniture. B2L is a business decision . If it's an area more than likely to be purchased to live keep the furniture as home ownership is an emotional decision. In both cases ensure the property is clean and tidy and presented in the best possible light given what you have. You don't need to spent a lot to get a home looking great .

On the whole, most people are lacking in imagination, and the photos really count, to give an ideal of size of rooms...which helps, even with published dimensions! So declutter, add a few pictures on the walls, perhaps a tablecloth, cushions, and pleasant bed covers etc. Especially if you want to attract non landlords as well.
If the buyer doesn't want the furniture, they can make it a condition of purchase that its removed before competion.

The long way:
1. Empty as many rooms of furniture as possible and scrap anything that is tatty. There may be a subjective case to furnish one bedroom if there is sufficient decent furniture to redeploy for staging purposes but only if it is good stuff.
Decorate the empty rooms in neutral colours. You really cannot go wrong with magnolia and white ceilings. I've only used Johnsons brilliant white vinyl silk on ceilings for the last 50 years which looks great but you do need good ceilings or it will highlight the imperfections.
I have the magnolia specially mixed to a lighter shade which brightens the whole property but doubles the cost of the paint.
2. Install daylight LED bulbs everywhere.
3. Clean all carpets using a truck mounted cleaner company. Don't even consider using anything else!
4. Less is more.
5. The above is not a royal decree and may be quite unsuitable in some circumstances, but it always for me.