Do the scholastic sects say that chan practice is something separate from themselves?

It is highly unlikely that Dogen's statement was critical of a scholastic Chan/Zen school, because there was none at the time. It is most likely that Dogen's statement was critical of (1) sectarian and (2) elitist positions of those who made 'following in the footstep of the Buddha' a sectarian matter. This include so-called masters whose awakening he doubted. At least, this is Steven Heine and Dale Wright's position, as I will quote later.

To come back on scholasticism, it never really took within the Chan tradition (contrary to Tendai etc.), probably partly because Chan is a late tradition. As such, Chan rested on established conclusions without having to establish them. These conclusions were established by other Buddhist schools by way of debates, contests, etc. Such conclusions are for instance the notion of 'one final vehicle', 'dharmakaya, all-pervasivenness & omniscience', 'the third turning of the wheel as teaching the final view', 'the Avatamsaka sutra being the first turning and the final view as well', and so forth. It is hard to find any Chan scholastic work outside Zongmi's and Yanshou's, and even these discuss Tenets more than they do the turnings of the Wheel. Moreover, Zongmi and his disciple are not always considered as 'Chan teachers' presumably because of it. In addition, when you study the lists of both 'Chan genres of texts' and 'Types of Chan masters', you find no reference to scholasticism.

Heine, Steven; Wright, Dale (2010), write in Zen Masters:

In Eihei kōroku 3.207, Dōgen criticizes Yunmen and the whole notion of the autonomy of a “Zen school,” which should not take priority over the universality of the Buddha Dharma: It is said that after [the Ancestor] Upagupta, there were five sects of Buddha Dharma during its decline in India. After Qingyuan and Nanyue, people took it upon themselves to establish the various styles of the five houses, which was an error made in China. Moreover, in the time of the ancient buddhas and founding ancestors, it was not possible to see or hear the Buddha Dharma designated as the “Zen school,” which has never actually existed. What is presently called the Zen school is not truly the Buddha Dharma.

And:

At times he is pansectarian in citing masters from all Chan schools, as well as nonsectarian in that he also denies the existence of an independent “Chan sect” altogether.

Is Dogen saying that zen is nothing separate from scholastic "sects"

If I can summarize the Butsudō from which your quote was taken, I think that what he's saying is that:

• The Teaching is personally transmitted from Dharma heir to Dharma heir ... the Buddha Dharma lives by being authentically transmitted.

• These were no sects in general (and no "Zen sect" in particular) in Shakyamuni Buddha's time, so it can't be right to say that there are different sects now. The Buddha and the Ancestors didn't talk about "the Zen sect", that kind of naming is wrong.

• Any method of training that's different from the methods used by the Buddha and the Ancestors is non-Buddhist.

• Not only shouldn't they be talking about "the Zen sect" people shouldn't be talking of different sects within Zen:

  My late Master, an Old Buddha, once ascended the Dharma seat and addressed his assembly, saying, “People nowadays just talk of there being separate traditions and customs, such as those of Ummon, Hōgen, Igyō, Rinzai, and Sōtō, but this is not the Buddha’s Teaching, nor is it what Ancestors and Masters say.”

• The disciples of Shakyamuni Buddha didn't want to miss any of his teaching, and neither should we.

  When Shakyamuni Buddha was in the world, His disciples were apprehensive lest they should differ from Him by even a hair, or lest they should miss even one jot within the hundreds of thousands of myriad points of His Teaching, for they wished to experience the joy of realizing the Truth which they had chosen as their inheritance, and they would not go against It. Thus, we should vow to seek and serve the Buddha and hear His Dharma over the course of many lives. Those who would deliberately go against the transforming Teaching that the World-honored One gave while He was in the world and give rise to sectarian names are not disciples of the Tathagata nor are they offspring of the Ancestral Masters.

Thee are three schools of Japanese Zen, however many Chans in China, stretch that thought through 1300 years, and your picture of Zen becomes less singular. In that it seemingly emphasises anapanasati over vipassana meditation, it is closer to the original texts imo - which itself is a reformatted memory of the Buddha's words. In any case, I think almost all buddhist monastic orders follow the Nikayas or Agamas, the Patimokkha and Vinaya, so there isn't much really to separate them.

Remember the concentration division of the Eightfold Noble Path consist of Right Effort, Right Mindfulness, Right Concentration.

The Noble Eightfold Noble Path did not specify a particular practice for obtaining mindful concentration as long as long as they are consistent with the teachings. You can obtain Right Concentration from Sitting Meditation "Zazen" or you can chant sutras, or you can do walking meditation. In fact for the Chan/ Zen practitioner they aspire to maintain Right Effort, Right Mindfulness and Right Concentration in every moment, in every thought. Some practices are easier than others to obtain concentration however, that's why sitting meditation is generally fundamental since it avoids distractions.

To summarize:

Right Effort

"And what, monks, is right effort?

[i] "There is the case where a monk generates desire, endeavors, activates persistence, upholds & exerts his intent for the sake of the non-arising of evil, unskillful qualities that have not yet arisen.

[ii] "He generates desire, endeavors, activates persistence, upholds & exerts his intent for the sake of the abandonment of evil, unskillful qualities that have arisen.

[iii] "He generates desire, endeavors, activates persistence, upholds & exerts his intent for the sake of the arising of skillful qualities that have not yet arisen.

[iv] "He generates desire, endeavors, activates persistence, upholds & exerts his intent for the maintenance, non-confusion, increase, plenitude, development, & culmination of skillful qualities that have arisen: This, monks, is called right effort."

http://www.accesstoinsight.org/ptf/dhamma/sacca/sacca4/samma-vayamo/index.html

Right Mindfulness

"One is mindful to abandon wrong view & to enter & remain in right view: This is one's right mindfulness...

"One is mindful to abandon wrong resolve & to enter & remain in right resolve: This is one's right mindfulness...

"One is mindful to abandon wrong speech & to enter & remain in right speech: This is one's right mindfulness...

"One is mindful to abandon wrong action & to enter & remain in right action: This is one's right mindfulness...

"One is mindful to abandon wrong livelihood & to enter & remain in right livelihood: This is one's right mindfulness..."

http://www.accesstoinsight.org/ptf/dhamma/sacca/sacca4/samma-sati/index.html

Right Concentration

"Now what, monks, is noble right concentration with its supports & requisite conditions? Any singleness of mind equipped with these seven factors — right view, right resolve, right speech, right action, right livelihood, right effort, & right mindfulness — is called noble right concentration with its supports & requisite conditions."

http://www.accesstoinsight.org/ptf/dhamma/sacca/sacca4/samma-samadhi/index.html

As long as the Noble Eightfold Path is being practiced, mindful concentration is possible and therefor the Jhanas also known as Chan or Zen is possible. Therefore the myriad practices of Buddhism lead to the same thing and there is no separation. You can scrub clean the floor and have mental concentration to experience Joy and Peace in your heart. Or you can chant sacred scriptures while performing complex mudras.

In theory, the Noble Eightfold Path must be practiced by every school of Buddhism and hence 'Chan/Zen' is practiced by every schools of Buddhism whatever their names maybe. However, as it tends to happen there are monks and schools who don't know what they are doing and spend their time debating philosophical concepts in the scriptures or running a temple like a business and not cultivating themselves and not practicing the Dharma. Even within the Chan/Seon/Zen school in China/Korea/Japan there are people who lost sight of what 'Zen' actually means and engaged in pointless philosophizing. As long you know how it work you can see the practices everywhere and not be confused by surface appearances.