The embedded assumption of that skill feature is that lodging (of some kind) is available.
That is made clear in the text of the description if one reads the whole description that you cited (PHB p. 130). Taking that one clause out of context — "always find a place to perform" — harms its contextual meaning. If you are in the middle of the forest, that ability will not create a tavern, an inn, nor a town in which to perform. (Granted, the birds and forest creatures may appreciate your song for its own sake). If you look at the last two sentences of the description,
In addition, your performance makes you something of a local figure.
When strangers recognize you in a town where you have performed, they
typically take a liking to you.
the initial condition (prerequisite?) is that there are other people around in a settlement of some size (town, village, city) to react to you favorably.
Are the woods really empty?
Per our discussion in comments, another way to look at this is to discuss with the player "who are you singing to here in the woods? Who is the audience?"
Is she singing to:
- Possible fey who are invisibly watching?
Any performer will usually perform for an audience. Rather than let this be an "on-off switch" there is a chance for some player input on how to make more use of this class feature in non-standard environments.
The PCs are in the woods on a very cold night. They have seen some animal life around, but it's cold and getting colder. The Bard sings or plays music and a bear comes near, interested in the music. It then lies down near the Bard and leaves a space for the Bard to snuggle right up to nice warm bear fur.
While that's a "cross-species" scenario, and you may want to call for a check to recall what kind of music calms beasts (Intelligence (Nature) check?); it's the kind of thing you and the Bard's player can work together on to get more out of the background feature.
It's not necessarily restricted to downtime activity
An adventure does not stop (although a given adventure day might end) simply by the need to take a long rest. Even with those nice lodgings that were earned by a performance, you can have encounters during the night such as:
- Thieves trying to steal the money you earned
- Groupies or fans trying to disturb your sleep (welcome or unwelcome)
- A local bard or musician visiting with a few thugs to run you off, or to scare you off, since you are seen as a threat to his income...
- Your ex wife (or ex husband) is living in this town, unbeknownst to you, and drops by later to talk to you about your running off for this life of adventure ... so much for that good night's sleep, regardless of how nice the tavern is.
Not necessarily down time, although if you do take downtime in a town or city, you can avoid spending money on that style of living.
Regarding old bards, and old bold bards
If you try this in an enemy war camp, you are likely to have to first make another kind of ability check so that you are given a chance to perform for them. Once on stage, "I'm dyin' up here" might not be as benign as at a tavern!
If you try this on a dragon, the same applies. Dragons have high wisdom and intelligence scores, and decent charisma scores. Depending upon how well you first establish a rapport with a dragon, a performance might keep you from being the main course. That's too situation dependent to turn "on" or "off" with this background feature, but it could make for a great interaction with a dragon!