Is Jaichand Rathod wrongly framed by historians as he helped Prithviraj in the Second Battle of Tarain?

Very likely, yes.

Contrary to what the educated classes say, it's pretty evident that Hindu rulers of the Late Classical/ Early Medieval were not only united in their hatred of the Mleccha for most part, but also extremely monocultural to a degree the modern Intellectual would bemoan as “Brahmanical Patriarchical Nazism”.

One can clearly see how spheres of Power had evolved on ethnic lines, with the Chahamana North, Gahadavala Madhyadesha, and Chalukya West- all of whom were usually related to each other in some way. Warfare along these States were primarily restricted to squabbles over border feudatories with the sort of Total War prevalent among Oriyas and Southerners being relatively rare.

In other words, the Northern Kings had, in response to the preceding 500 years of near-constant warfare with the Mleccha- created a system that explicitly prized the Security of their Peoples, restricted inter-Bharata violence, diverted resources from centralised Total War to a more defensive and diffusive system, and - very importantly- led to a social milieu where Martial Honour made it obligatory to fight the Mleccha.

Now we know that there used to be great libraries detailing the history of - what we'd call today- middle “Rajput” kingdoms (the terms are anachronisms as my usage of the “real” names of these Dynasties demonstrate) at Varanasi, Vallabhi, Bhita and other places where scholars gathered… But apparently they never existed since the Intellectuals say that Hindus arose from Hitler's bunker in 1945 CE.

Anyway…

Jayachandra Gahadavala, in the late 12th Century, HAD to march in defence of the Chahamana.

Not only because Maharaja Prithviraja was his own son-in-law but because prevailing societal law over all the North and West DEMANDED the prevailing Sovereigns loathe Mlecchas and kill them whenever they get the chance. This was, by no means, a new phenomenon. In the words of the Mlecchas themselves, they had no fiercer foe than the Gurjara Pratihara Emperors.

(Trivia: “Pratihara” literally means “Chamberlain”. Why would an Imperial House claim descent from a family of mere Chamberlains serving someone else? The answer is obvious but an intellectual will never get it.)

At any rate, the usual Intellectual line- parroted from Mleccha hearsay or unreliable folk tales- that Maharaja Jayachandra was a traitor to Dharma holds no water. Even if the latter disliked his Chahamana kin, he had to march nonetheless or his own feudal lords would've either deposed him or used his perceived cowardice as an excuse to either revel or crown a pretender. As such, we observe that the period following the collapse of the Chahamana was followed by bloody warfare across the North.

Jayachandra himself fell within a single year of the sacking of Delhi, at a location that'd come nominally under Chahamana control. Implying that he was, in no manner, neglecting his duties to his Throne and possibly striving for revenge against the Mlecchas for the wrongs they're always committing.

The likeliest reason for Maharaja Prithviraja's defeat, in my opinion, is his own merciful nature which not only let the Mlecchas escape their deserved retribution but would've also provided them with vital eyewitness intelligence regarding Hindu preparedness and strategy. A more hard-headed ruler like Maharaja Anangabhimadeva III Chodaganga wouldn't have had a second's thought over brutally torturing some captured Mleccha general to death.

As Uncle Janardhan Stalinacharya said: “No Turaka Mleccha to fight in 1191 CE, no question of discussing who betrayed whom in 2018 CE.”

BTW I'm very skeptical that the Gahadavala were Rathores. Rao Naths historically came out prominence over what we'd call Solanki-Chahamana borders, possibly as a direct result of the fall of Delhi. The oldest known Rathore King is a certain Rao Simha who took over Barmar around 20–25 years after Maharaja Prithviraja's death.