Why this reading doesn't exist in our modern Moshaf?
This is a known reading of the qur'an that is no more accepted since the compilation of the moshaf al-Imam by 'Uthman ibn 'Affan which can be regarded as a canonical copy. As it neither is conform to the scripture nor the recitation on which most of the qurraa' of 'Uthman's committee among the sahabah agreed upon.
And the three main conditions for accepting a qira'a are for example explained in ibn al-Jazari's ابن الجزري Teebat an-Nashr طيبة النشر في القراءات العشر (Source):
In the following I'm translating from Arabic language, to the best of my knowledge. I'm not a professional translator. Please take these translations with the necessary care.
14 - فكل ما وافق وجه نحو ... وكان للرسم احتمالا يحوي
All which corresponds to a (possible Arabic) grammar case ... and was (potentially) conform to the rasm (scripture)
15 - وصح إسنادا هو القرآن ... فهذه الثلاثة الأركان
And has a sound/authentic (Sahih) sanad (narrator chain) is the qur'an ... these three are the arkaan (pillars)
16 - وحيثما يختل ركن أثبت ... شذوذه لو أنه في السبعة
And wherever one of these pillars is broken it is proven ... that its an anomaly (shudhood) even if it were among the seven (qira'at)
Even if there were more reliable or authentic narrations the second pilar is broken here.
Therefore it is considered as an explanatory reading which falls under the category of qira'a shadha. Further I'd even suppose (needs to be checked) that if we check the origin and original use of these two verbs you may find that each of them was the preferred term used to express going/walking by different Arabic tribes.
The former existence of this reading can be confirmed
It is also known that 'Umar ibn al-Khattab was always reciting it "فامْضُوا إلى ذِكْرِ اللَّهِ" while Ubay ibn Ka' ab rectited "فَاسْعَوْا إِلَى ذِكْرِ اللَّهِ" some scholars commented this choice or reading of 'Umar saying something like:
where there not the reading (qira'a) فامْضُوا إلى ذِكْرِ اللَّهِ people would come to the prayer runnig.
The term used for this running in the quote I've heard and read was a third synonym of these two terms: هرولة
Imam at-Tabari in his Tafsir (see for example here in Arabic) actually quoted several narrations of which some have a high level of authenticity that confirm 'Umar's choice.
For example in the following chain all narrators are trustworthy and appear in at least Sahih Muslim:
'Abdulhameed ibn Bayaan as-Sukkary عَبْدُ الْحَمِيدِ بْنُ بَيَانٍ السُّكَّرِيُّ -> Sufyan سُفْيَانُ (ibn 'Uyyanyah) -> az-Zuhri الزُّهْرِيِّ -> Salim سَالِمٍ ibn 'Abdullah-> His father أَبِيهِ (Abdullah ibn 'Umar)
But al-Moshaf al-Imam was only compiled after his death. So we can confirm that this qira'a existed and was practiced, but due to the canonization is regarded as invalid in order to avoid fitna in the Ummah which was the major reason for compiling al-Moshaf al-Imam.
A deeper look in the Arabic terms
This is because the verb
سعى، يسعى، اسع
or the corresponding noun السعي is often used even in the qur'an and the terminology of fiqh to express a faster way of walking or moving forward like as-Sa'y السعي between Safa and Marwa during Hajj:
وَجَاءَ رَجُلٌ مِّنْ أَقْصَى الْمَدِينَةِ يَسْعَى...
And a man came from the farthest end of the city, running. ... (28:20)
وَجَاءَ مِنْ أَقْصَى الْمَدِينَةِ رَجُلٌ يَسْعَىٰ قَالَ يَا قَوْمِ اتَّبِعُوا الْمُرْسَلِينَ
And there came from the farthest end of the city a man, running. He said, "O my people, follow the messengers. (36:20)
In fact the verbs:
have in common being a synonym of
ذهب (to go)
See also What were the criteria for validating a qira'a?