This is false of the UK, where since the Glorious Revolution the Parliament, and the executive that is responsible to it, handles all legislation.
This is misleading of the UK, because it elides (i) the power of a Prime Minister to advise the monarch to appoint new peers, and (ii) the effect of the Parliament Acts, which mean that the House of Lords plays a procedural but not a determinative role, and (iii) the effects of the Blair-era changes to the House of Lords that mean it is an overwhelmingly appointed chamber.
Well, except for the bits about "most of the day to day legislation" and the fact that it would be a quite misleading description of the UK system.