After update 2024.6.1 problems with GUI

### The problem

GUI sometimes without content


the same situation with editor - opening existing project = empty editor


### Which version of ESPHome has the issue?

2024.6.1

### What type of installation are you using?

Home Assistant Add-on

### Which version of Home Assistant has the issue?

2024.2.5 (no time for tests using current, but it does not look like HA core related)

### What platform are you using?

ESP8266

### Board

not platfom dependent

### Component causing the issue

not dependent

### Example YAML snippet

```yaml
not dependent
```

### Anything in the logs that might be useful for us?

```txt
not dependent
```

### Additional information

I have never seen a similar situation in any of the older versions of ESPHome (2024.6.0 wasn't used as update was from 2024.5.5)

Browser cache cleared (as usual when any problems with HA GUI), but doesn't help.

Maybe related to
https://github.com/esphome/esphome/pull/6944

Start by reproducing the empty GUI and editor with ESPHome 2024.6.1 as a Home Assistant Add-on, using the screenshots and the report that clearing the browser cache did not help. Review the possibly related pull request #6944 and compare against 2024.5.5; done means existing projects and the GUI consistently display their content.

Written by the indexing model from the issue text.