Is it better for woman to go out to the Eid prayers or to stay at home?

1. There is no prohibition for women to attend the Mosque in Islam.
2. There is discouragement for women to attend the Mosque in Islam because Nabi (Sallallahi Alaihe Wassallam) advised a Sahabi woman that her Salah at her home is superior to her Salah:

• In Masjidun-Nawabi
• Behind Nabi (Sallallahi Alaihe Wassallam) as the Imam
  

Now apply this to her local Mosque.

To understand this simple concept, picture this:

1. There is no prohibition for men to pray individually
   
2. BUT they are discouraged to pray individually and enocuraged to pray with congregation

So anyone who loves the Sunnah should grasp both concepts.

Your argument that women are already outdoors is discussed by Mufti Tariq Saheb (HA) and you will find it in original Urdu and translation here:

What is the ruling for women attending Mosques?

Same concept applies for Eid Salah within the Hanafi Madhab as per the practise of many of the Salaf. There are others who disagree and state that women should go to Eid Salah and the issue of praying in the Mosque is separate and distinct.

Hanafi evidence from the Qur'aan and Sunnah is here

www.muftisays.com/blog/Muadh_Khan/3548_07-02-2014/are-deo...

The other side state their evidence that women should be encouraged to attend Eid Salah.

Let me tell you ground reality.

I was teaching in a Masjid in my class. A Sister knocks and walks into my Class with a whole bunch of Non-Muslims (Dawah Tour) smiling and joking and without proper Hijab.

The Mosque decided to show my class to the visiting Non-Muslims, I don't have a fundamental problem with the approach. BUT:

1. She was without Hijab
2. She was interacting with men who were her Ghair-Mahram
3. It is against rules and decency to disrupt a class

I said nothing BUT the Non-Muslims apologised and said, "O! We didn't mean to disrupt you guys and apologised". She was so unprofessional and STUPID that she didn't even get the point and just kept similing...

This is how STUPID this issue has become. In the same Masjid there are Sisters who are following sunnah and professionals who could have handled the same job better BUT because they are not Deobandees, this (Sister of a Deobandi) was preferred with no Hijab, no manners and half knowledge of Islam.

I had to stop my lesson and switch to basics about Islam in a manner which didn't feel forced so I went "O welcome, we were just talking about....Join the conversation...."

This Mosque is Hanafi and for years DID NOT allow women to pray and now they have women running out doing open days without any restrictions etc.

In another (Deobandi) Mosque which DOES NOT allow women to pray every few months they have Charity collections right in the Parking lot where women (covered, uncovered) are selling things, raising money talking to men openly.

I parked the car and a Sister came smiling to me, "Brother, would you like a Samosa and a Tea"....I am like what? Who are you? What's happening here!!!

The point is that Deobandees have really lost the plot!

They have gone from undue restriction to no restrictions at all.

Go to most Deobandi Mosques in Februrary on Open Days and you will see that it is free for all.  Salafi Mosques don't have this sort of problem. Go back to my posts on SF years ago and I predicted this will happen.

Its a mess...All because our Ulama have never tackled the situation and never trained the local communicities on how to handle this issue.

I don't know Mount Pleasant but agree with your thinking. Women can sell Samosas but cannot pray!

Makes no sense.