Queen Victoria ignored or banished Princess Alice from visiting Britain and therefore her young family couldn't go either
That statement does not appear to be supported by the historical evidence. Queen Victoria was not in a position to banish a married daughter from visiting Britain although she could be discouraging and refuse to help financially - as I said earlier, Vicky was sometimes put in the same position, so it wasn't just aimed at Alice. Alice in fact was with Queen Victoria often during her married life, and in England where on a number of occasions she wrote to the Queen from Buckingham Palace (presumably to QV in Windsor or Balmoral or Osborne) or Sandringham, suggesting her mother's hospitality was not always limited either. From references in Queen Victoria's letters, and from Alice's own letters to Queen Victoria, it is possible to see that she visited England in 1863, 1864, 1865, 1868, 1871, 1872, 1873, 1875, 1876 and 1877. It is unlikely that she visited in 1866-7, during the Austro-Prussian war and its aftermath, or in 1870 during the Franco-Prussian war. I'm not sure what happened in 1869, 1874 or 1878 - I haven't found any references, but she might well have visited Britain. At any rate, Alix of Hesse could well have travelled to Britain four out of the six years of her life until her mother died and would have undoubtedly seen Queen Victoria on many occasions during that period. Afterwards, Queen Victoria unquestionably felt she had taken Alice's place as a mother to the motheless Hesse-Darmstadt children and they visited her very frequently.
I agree with Eddie that she did not feel guilty about her treatment over Alice - she certainly felt sorrow at her death, but in her correspondence whenever she complained of her behaviour she seemed fully convinced that she was right to do so. Problems seemed to have occurred particularly in 1867, when QV felt Alice had criticised Princess Helena for not travelling to Germany with her husband Prince Christian, in 1868, when she thought Vicky and Alice encouraged Prince Albrecht of Prussia to court Princess Louise against both her and Louise's wishes, and in 1872, when the Queen thought Alice had spread a rumour that Louise had criticised Francis Knollys as an unsuitable companion for the Prince of Wales (Louise said Alice herself was responsible for the rumour). It all sounds like the usual family rows - and they did blow over. Queen Victoria could be controlling and ungenerous, but she could be kind and sympathetic also, and like most of her children, Alice was sometimes in favour and sometimes not. But when she died, QV was genuinely upset and determined to do right by her children.