Should throw in a UV bath as well - to kill any airborne spores/other critters.

Absolutely, there are plenty of climates where HRV (aka MVHR) is quite sufficient. In inland areas (most of the US, and a fair chunk of Europe), the temperature difference between summer and winter is enough that a HRV is unlikely to be sufficient. Somewhere like the UK or somewhere near the equator and high altitude, a HRV system probably will be absolutely fine.

In the UK, the outside air dew point in the depths of winter is likely to be between -5 and 15 °C, and in summer likely between 10 and 25 °C. That might be a little dry at the driest, and a little sticky on a couple of days in the summer, but it's perfectly reasonable to just bring that air into the house without messing with the humidity. Most UK houses are too humid and don't have enough ventilation.

However, as soon as you start wanting to cool a house because the outside air is too hot, then that outside air will basically have 100% relative humidity when it is cooled to your house temperature, and it will probably have condensed out a load of its moisture onto whatever was cooling it, so you absolutely do need to mess with the humidity.

In my climate the problem is air being too dry inside during winter.

Outside air is normal humidity 50-80% RH at -10C. Inside it will be 20% and way too dry.

Ventilation, heat recovery and moisture recovery would be ideal.

During summer we rarely cook with and AC and windows are open most of the time.

I think the issue is that when the heat exchange occurs one way, the HRV is effectively a condenser.

Moisture builds on the heat exchange surfaces of the heat exchanger, particulates, spores, etc. in the air deposit nutrients and viable life in the crud, it thrives and multiples right into the incoming airstream for your residence.

When the window is opened the air mixes without a surface for condensation and there's much less immediate deposition so there's not an enriched medium to sustain accelerated growth entirely in the air stream, but it still probably occurs to some extent in a more diffuse manner that produces a significantly lower ambient spore/toxin/particulate load

There could be some issue with condensation. The moisture from the indoor air could get trapped and leak.

How does that remove the condensation though? That's the cause of the mold problem.

Edit: looks like somehow it moves moisture between the exhaust/intake somehow... not just heat.

What does that system cost? There is no pricing information on the website.