Difficult to tell from what you have written what the problem is, but perhaps the information from the EU site might help:

Your non-EU spouse and children's residence rights in the EU - Your Europe

You are an EU citizen moving to another EU country to live, work or study? Your spouse, children and grandchildren can you, even if they are not EU nationals.

europa.eu

It depends a bit on where you are located within the EU as the exact requirements vary quite a bit from one state to the next. I only note that you mention that your wife was rejected for an entry visa. The countries I am familiar with within the EU tend to waive the need for the non-EU spouse (or close family member) to have a visa at all and may go directly to issuing them a residence permit when coming to their EU national spouse in the country.

 

I live in Finland. To apply for the "Residence card of a family member of an EU citizen in Finland" the Migri website states:
"The application can only be submitted in Finland. You must live in Finland when you submit the application. You need to be residing most of the time in Finland."

Residence card for family | Maahanmuuttovirasto

So, from what I understand, my wife needs an entry visa to me here. Therefore, we applied for a visa as an EU family member.

 

I live in Finland. To apply for the "Residence card of a family member of an EU citizen in Finland" the Migri website states:
"The application can only be submitted in Finland. You must live in Finland when you submit the application. You need to be residing most of the time in Finland."

Residence card for family | Maahanmuuttovirasto

So, from what I understand, my wife needs an entry visa to me here. Therefore, we applied for a visa as an EU family member.

 

I took a look at the site you mention, and they say (under the heading that mentions that the person must have a valid port) the following:

You need to have a valid port. You can come to Finland with a visa or visa-free.

I think what they're indicating is that it may depend on where your wife is from (i.e. what port she holds). If your wife is from a country that does not need a visa (or only needs the stamp-in-the-port "Schengen visa") then she may only need to prove that she is living with you.

If your wife is from a country that requires a visa, even for a short, "tourist stay" then she should be OK with a short-term tourist visa. (This is similar to how things work in - though in the non-EU partner must apply for the residence card within 90 days of their arrival in . There doesn't seem to be a limit like this in the Finnish procedure.

 

I took a look at the site you mention, and they say (under the heading that mentions that the person must have a valid port) the following:

I think what they're indicating is that it may depend on where your wife is from (i.e. what port she holds). If your wife is from a country that does not need a visa (or only needs the stamp-in-the-port "Schengen visa") then she may only need to prove that she is living with you.

If your wife is from a country that requires a visa, even for a short, "tourist stay" then she should be OK with a short-term tourist visa. (This is similar to how things work in - though in the non-EU partner must apply for the residence card within 90 days of their arrival in . There doesn't seem to be a limit like this in the Finnish procedure.

My wife is from a non-EU country and needs a short-stay Schengen visa to enter Finland. The embassy representing Schengen visa applications for Finland in her country is the Netherlands. We came across information regarding "Visa facilitation for family of EU/EEA or Swiss nationals" on the following website:

I am a family member of a national of an EU/EEA member state or Switzerland. Can I get visa facilitation for the Netherlands? | NetherlandsWorldwide

If you are a family member of a national of an EU member state, an EEA member state (Iceland, Liechtenstein or Norway) or Switzerland, check whether you're eligible for free Schengen visa facilitation for your trip to the Netherlands.

www.netherlandsworldwide.nl

On the above page, the third condition states: "You are traveling to or with your family member." My wife's intention was to travel to Finland to me. However, based on the rejection, it seems that I am also required to travel with her. However, in her visa application, under the Purpose(s) of the journey section, we only selected "Family of EU/EEA." I now realize we should have explained her situation more clearly, stating that her intention was to her husband in Finland.

https://www.netherlandsworldwide.nl/binaries/content/assets/pdfs-engels/checklist-visa-family--eu-eea-swiss-nationals-28-12-2022.pdf

In the checklist of documents provided in the link above, there is no mention of financial requirements or the need for an apartment for the Schengen visa application for a family member of an EU/EEA national. However, I do have enough money in my to cover her stay and also have an apartment.