How to the image date/ metadata used when using the .median filter in GEE?

Question

I am processing LST, NDVI, NDBI, & UHI data in a 1 year period. I used the .median filter function for filtering the dates.

My question is, how do I know what images to use? I've tried the following (https://developers.google.com/earth-engine/guides/image_info) but it only works with the .first filter.

Here's my complete code:

    // INPUTS` **********************************************************************************
    
    // AOI
    var startDate = '2023-01-01'
    var endDate = '2023-12-31'
    
    // *****************************************************************************************
    
    // Applies scaling factors.
    function applyScaleFactors(image) {
    var opticalBands = image.select('SR_B.').multiply(0.0000275).add(-0.2);
    var thermalBands = image.select('ST_B.*').multiply(0.00341802).add(149.0);
    return image.addBands(opticalBands, null, true)
              .addBands(thermalBands, null, true);
    }
    
    //cloud mask
    function maskL8sr(col) {
    // Bits 3 and 5 are cloud shadow and cloud, respectively.
    var cloudShadowBitMask = (1 << 3);
    var cloudsBitMask = (1 << 5);
    // Get the pixel QA band.
    var qa = col.select('QA_PIXEL');
    // Both flags should be set to zero, indicating clear conditions.
    var mask = qa.bitwiseAnd(cloudShadowBitMask).eq(0)
                 .and(qa.bitwiseAnd(cloudsBitMask).eq(0));
    return col.updateMask(mask);
    }
    
    // Filter the collection, first by the aoi, and then by date.
    var image = ee.ImageCollection('LANDSAT/LC08/C02/T1_L2')
    .filterDate(startDate, endDate)
    .filterBounds(aoi)
    .map(applyScaleFactors)
    .map(maskL8sr)
    .median();
    
    
    var visualization = {
    bands: ['SR_B4', 'SR_B3', 'SR_B2'],
    min: 0.0,
    max: 0.3,
    };
    
    Map.addLayer(image, visualization, 'True Color (432)', false);
    
    // NDVI
    var ndvi  = image.normalizedDifference(['SR_B5', 'SR_B4']).rename('NDVI')
    Map.addLayer(ndvi, {min:-1, max:1, palette: ['blue', 'white', 'green']}, 'ndvi', false)
    
    // ndvi statistics
    var ndvi_min = ee.Number(ndvi.reduceRegion({
    reducer: ee.Reducer.min(),
    geometry: aoi,
    scale: 30,
    maxPixels: 1e9
    }).values().get(0))
    
    var ndvi_max = ee.Number(ndvi.reduceRegion({
    reducer: ee.Reducer.max(),
    geometry: aoi,
    scale: 30,
    maxPixels: 1e9
    }).values().get(0))
    
    
    // fraction of veg
    var fv = (ndvi.subtract(ndvi_min).divide(ndvi_max.subtract(ndvi_min))).pow(ee.Number(2))
          .rename('FV')
    
    
    var em = fv.multiply(ee.Number(0.004)).add(ee.Number(0.986)).rename('EM')
    
    var thermal = image.select('ST_B10').rename('thermal')
    
    var lst = thermal.expression(
    '(tb / (1 + (0.00115 * (tb/0.48359547432)) * log(em))) - 273.15',
    {'tb':thermal.select('thermal'),
    'em': em}).rename('LST')
    
    var lst_vis = {
    min: 25,
    max: 50,
    palette: [
    '040274', '040281', '0502a3', '0502b8', '0502ce', '0502e6',
    '0602ff', '235cb1', '307ef3', '269db1', '30c8e2', '32d3ef',
    '3be285', '3ff38f', '86e26f', '3ae237', 'b5e22e', 'd6e21f',
    'fff705', 'ffd611', 'ffb613', 'ff8b13', 'ff6e08', 'ff500d',
    'ff0000', 'de0101', 'c21301', 'a71001', '911003']
    }
    
    Map.addLayer(lst, lst_vis, 'LST AOI')
    Map.centerObject(aoi, 10)
    
    // Urban Heat Island ***********************************************************************
    
    //1. Normalized UHI
    
    var lst_mean = ee.Number(lst.reduceRegion({
    reducer: ee.Reducer.mean(),
    geometry: aoi,
    scale: 30,
    maxPixels: 1e9
    }).values().get(0))
    
    
    var lst_std = ee.Number(lst.reduceRegion({
    reducer: ee.Reducer.stdDev(),
    geometry: aoi,
    scale: 30,
    maxPixels: 1e9
    }).values().get(0))
    
    
    
    print('Mean LST in AOI', lst_mean)
    print('STD LST in AOI', lst_std)
    
    
    var uhi = lst.subtract(lst_mean).divide(lst_std).rename('UHI')
    
    var uhi_vis = {
    min: -4,
    max: 4,
    palette:['313695', '74add1', 'fed976', 'feb24c', 'fd8d3c', 'fc4e2a', 'e31a1c',
    'b10026']
    }
    Map.addLayer(uhi, uhi_vis, 'UHI AOI')
    
    // Urban Thermal Field variance Index (UTFVI)
    
    var utfvi = lst.subtract(lst_mean).divide(lst).rename('UTFVI')
    var utfvi_vis = {
    min: -1,
    max: 0.3,
    palette:['313695', '74add1', 'fed976', 'feb24c', 'fd8d3c', 'fc4e2a', 'e31a1c',
    'b10026']
    }
    Map.addLayer(utfvi, utfvi_vis, 'UTFVI AOI')
    
    
    // *****************************************************************************************
    
    // NDBI
    var ndbi  = image.normalizedDifference(['SR_B6', 'SR_B5']).rename('NDBI')
    Map.addLayer(ndvi, {min:-1, max:1, palette: ['white', 'orange', 'red']}, 'ndbi', false)
    
    // Display all metadata.
    print('All metadata:', image);
    
    // Get a specific metadata property.
    var cloudiness = image.get('CLOUD_COVER');
    print('CLOUD_COVER:', cloudiness);  // ee.Number
    
    // Get version number (ingestion timestamp as microseconds since Unix epoch).
    var version = image.get('system:version');
    print('Version:', version);  // ee.Number
    print('Version (as ingestion date):',
          ee.Date(ee.Number(version).divide(1000)));  // ee.Date
    
    // Get the timestamp and convert it to a date.
    var date = ee.Date(image.get('system:time_start'));
    print('Timestamp:', date);  // ee.Date
    
    // Donwload Param
    var params = {
      name: 'ndvi',
      crs: 'EPSG:4326',
      scale: 30,
      region: aoi,
      filePerBand: false,
      format: 'GeoTIFF' 
    };