Depends - the problem with heat pumps is when you need them the most they don't work. If it never gets below -10c (exact temperature needs more study, could be as low as -25) where you live they are fine - but that implies you live in an area where you don't get many cold days and so the expense isn't worth it (it also implies you live where it gets hot in sumner so you want ac anyway and the marginal additional cost makes it worth it again). If you live in an area where it gets colder you need additonal backup heat that can cover those really cold days and so you may as well run that system only.

I think unless you're in an area dominated by cooling needs, an optimally sized heat pump system will not cover 100% of heating needs. It would make sense to make it smaller and use a backup resistive heater for rare very cold events.

cooling is more important but not by much. however the real problem is temperature delta: 100f-70f is 30 degrees, 70f-10f is 60 degrees. If you size a system for cooling it can't make up.

of course things arenot actually linear on temperature but as a rough estimate it gets the point across.

Efficiency allows you to use less solar panels, but more solar panels are cheaper than a heat pump. I think the ratio is about 5:1 at this point and widening.

To be concrete, I'm told that recently in the US a certain 34000btu/hour (10kW) output heat pump consuming up to 14A at 220V at the compressor (3kW) cost US$2700 installed, which is 27¢ per peak watt of output. But https://www.solarserver.de/photovoltaik-preis-pv-modul-preis... gives a price of €0.055 per peak watt (US$0.065/Wp) for low-cost solar panels. So the heat pump costs, in some sense, 4.2 times as much as the solar panels.

But the heat pump doesn't save you 10kW over resistive heating when it's running full-tilt. It saves you 10-3 = 7kW. So it costs 39¢ per watt of saved energy, which is 6 times as much as the solar panels.

In some simplified theoretical sense, if you decide you need another 10kW of heating for your house, you could spend US$2700 on this heat pump, and also buy 3000 Wp of solar panels to power it, costing US$194, for a total cost of US$2894. Or you could buy 10000 Wp of solar panels, costing US$645, and a resistive wire, costing US$10, for a total cost of US$655. US$655 is almost five times cheaper than US$2894. (4.4 times cheaper.)

There are a lot of factors that this simplified cost estimate overlooks; for example:

• Maybe you need to run the heater 16 hours a day but you only get sunlight 7 hours a day, either because it's winter in Norway, or because there are tall pine trees that shade your property most of the day, and you can't put the panels up on the trees. So maybe in some sense one watt of peak heater output is worth 2.3 watts of peak solar panel output. Or maybe it's the other way around, where your house only needs active heating during a few hours at night, so one watt of peak heater output is only worth 0.43 watts of peak solar panel output.

• The prices are in different countries. Solar panels are more expensive in the US, even wholesale.

• US$2700 is the retail price of the heat pump, including installation and warranty, and 6.5¢/Wp is the wholesale price of low-cost solar panels with no warranty ("Minderleistungs-Solarmodule, B-Ware, Insolvenzware, Gebrauchtmodule, PV-Module mit eingeschränkter oder ohne Garantie, die in der Regel auch keine Bankability besitzen.") Even in Europe the retail price of solar panels is three or four times this.

• Driving a resistive heating element from solar panels is considerably easier than driving a heat pump from solar panels; adapting a heating element to run on lower voltage is just a matter of connecting more wires to the middle of it, while adapting a heat pump to run on lower voltage may involve redesigning the whole power supply board or even rewinding the motor. Which is in a hermetically sealed refrigerant circuit, by the way, which you'd have to reseal. In practice, you'd just buy an inverter, but a 3000-watt inverter is expensive.

• As you said, for sensible-heat thermal storage, the heat pump craps out at about 50° or 60°, while any garden-variety resistive heating element (plus a lot of crappy improvised ones) will be just fine at 600° or 700°. That means you need ten times as much thermal mass for the same amount of storage. Sand is dirt cheap, but once you get into the tens of tonnes, even dirt isn't really cheap.

Despite such complications, I still think that pair of numbers is a useful summary of the situation: the heat pump costs 39¢ per watt saved, while the solar panel costs 6.5¢ per watt produced.