Schools feel somewhat behooved to do a better job.The speaker was a sports writer, she reports, named Kevin Blackistone. I first thought this was the stylistic overreaching of a sports writer, but the construction gets over 59,000 g-hits, or "using the google" as Mrs. Verb (not her real name) likes to put it. In those hits, it's often used to set up a reply or response to incorrect information, so the typical meaning looks like 'feel compelled' or something: "I feel behooved to let your readers know … ." But Blackistone's example would have to mean 'be encouraged, feel more inclined' or something. To feel behooved shows up in lots of formal contexts, including an article in the journal College English.
I don't find comments on this particular construction in a quick look around, but the Oxford English Dictionary gives this:
5. a. Used, owing to confusion between the accusative and nominative (see first two quots.), as a personal verb: To be under obligation (to do); = must needs, ought, have. Of northern origin, and since 1500 only Scotch.From there, I suppose you could get to the feel + past participle construction. But I'm not sure how to think about the resulting construction — seems like a stretch to call that behooved an adjective and semantically it seems almost like a passive. I guess syntax folks have stories about to feel disappointed and such, and if my insurance covered the drugs to control syntax allergies, I would ponder the point.
[c1340 HAMPOLE Prose Tr. (1866) 5 e nam of Ihesu es helefull and nedys by-houys be lufed of all. c1386 CHAUCER Pars. T. 557 A servaunt of God bihoveth nought to chide.] c1400 Apol. Loll. 31 Swelk men be howuen tak hede. 1475 CAXTON Jason 76 The..craft that he behoueth to obserue and kepe. 1549 Compl. Scot. xv. 131 We behufit fyrst to reueil it. 1637 GILLESPIE Eng. Pop. Cerem. II. ix. 52 He behooved to offend the Iewes. 1759 ROBERTSON Hist. Scot. II. VIII. 45 They behoved to esteem them traitors. 1832 SIR W. HAMILTON Disc. (1853) 101 He behoved ... clearly to determine the value of the principal terms
Whatever it is, it's completely impossible for me, and the missus was downright taken aback by it.