Verse 1.2.8 has already stated that varnasrama duties (dharmah svanusthitah) are wasted endeavor. And even jnana and yoga, devoid of bhakti, are wasted endeavor.

sreyah-srtim bhaktim udasya te vibho klisyanti ye kevala-bodha-labdhaye
tesam asau klesala eva sisyate nanyad yatha sthula-tusavaghatinam

My dear Lord, devotional service unto You is the best path for self-realization. If someone gives up that path and engages in the cultivation of speculative knowledge, he will simply undergo a troublesome process and will not achieve his desired result. As a person who beats an empty husk of wheat cannot get grain, one who simply speculates cannot achieve self-realization. His only gain is trouble. SB 10.14.4

naiskarmyam apy acyuta-bhava-varjitam na sobhate jnanam alam niranjanam
kutah punah sasvad abhadram isvara na carpitam karma yad apy akaranam

Knowledge of self-realization, even though free from all material affinity, does not look well if devoid of a conception of the Infallible [God]. What, then, is the use of fruitive activities, which are naturally painful from the very beginning and transient by nature, if they are not utilized for the devotional service of the Lord? SB 1.5.12

pureha bhuman bahavo ’pi yoginas tvad-arpiteha nija-karma-labdhaya
vibudhya bhaktyaiva kathopanitaya prapedire ’njo ’cyuta te gatim param

O almighty Lord, in the past many yogis in this world achieved the platform of devotional service by offering all their endeavors unto You and faithfully carrying out their prescribed duties. Through such devotional service, perfected by the processes of hearing and chanting about You, they came to understand You, O infallible one, and could easily surrender to You and achieve Your supreme abode. SB 10.14.5

From this chapter it has already been understood that only by pure bhakti, unmixed with karma, jnana or yoga, the mind becomes satisfied.

A doubt arises. Is there not some worry in giving up jnana and yoga? And in not doing nitya-karmas, is there not the calamity of great sin in omission of action, and ending up in hell? This verse answers. Though jnana and yoga have some attraction, they depend on getting their results by performance of bhakti. But bhakti gives its results without the assistance of jnana or yoga at all. Therefore if one pleases the Lord by bhakti only, that is the perfection of dharma. The person who does not get for the most part perfection even though performing all the rules and regulations of karma attains perfection directly through bhakti, even though he does not perform the karmas. This is understood from the Lord’s own words:

yat karmabhir yat tapasa jnana-vairagyatas ca yat |
yogena dana-dharmena sreyobhir itarair api ||
sarvam mad-bhakti-yogena mad-bhakto labhate ’njasa |
svargapavargam mad-dhama kathancid yadi vanchati ||

Everything that can be achieved by fruitive activities, penance, knowledge, detachment, mystic yoga, charity, religious duties and all other means of perfecting life is easily achieved by my devotee through loving service unto me. If somehow or other my devotee desires promotion to heaven, liberation, or residence in my abode, he easily achieves such benedictions. SB 11.20.32.-33

Thus the idea that the devotee incurs sin by not performing karmas is discarded.

If dharma is perfected by bhakti, will the performer of dharma still obtain the results of dharma? Yes. If one performs dharma with material desires, one will attain those desires, but if one performs dharma without material desires, one will not attain material results. Gopala-tapani sruti says:

bhaktir asya bhajanam | tad ihamutropadhi-nairasyenaivamusmin manah- kalpanam | etad eva ca naiskarmyam ||

Bhakti is worship of the Lord, concentrating the mind on Him, renouncing all material desires for enjoyment (upadhi) in this world and the next. It destroys all karmas. Gopala-tapani Upanisad 1.15

According to the following verse, since all dharmas are perfected by bhakti, the necessity of the devotees performing dharma is rejected.

yatha taror mula-nisecanena trpyanti tat-skandha-bhujopasakhah
pranopaharac ca yathendriyanam tathaiva sarvarhanam acyutejya

As pouring water on the root of a tree energizes the trunk, branches, twigs and everything else, and as supplying food to the stomach enlivens the senses and limbs of the body, simply worshiping the Supreme Personality of Godhead through devotional service automatically satisfies the demigods, who are parts of that Supreme Personality. SB 4.31.14

The Lord himself says:

tavat karmani kurvita na nirvidyeta yavata |
mat-katha-sravanadau va sraddha yavan na jayate ||

As long as one is not satiated by fruitive activity and has not awakened his taste for devotional service by sravanam kirtanam visnoh one has to act according to the regulative principles of the Vedic injunctions. SB 11.20.9

ajnayaivam gunan dosan mayadistan api svakan
dharman santyajya yah sarvan mam bhajeta sa tu sattamah

Having taken complete shelter at My lotus feet, however, a saintly person ultimately renounces such ordinary religious duties and worships Me alone. He is thus considered to be the best among all living entities. SB 11.11.32

And in the Gita the Lord says:

sarva-dharman parityajya mam ekam saranam vraja

Give up all dharma and just surrender to me. BG 18.66

Based on these verses, the meaning of the verse yatha toror mula-nisecanena is as follows. Just as worship of the Lord takes the place of worshiping all devatas, and satisfaction of the Lord is the perfection of performance of varnasrama duties, when the Lord is satisfied with his worship, the goal of performing one’s duties including worship of devatas is automatically achieved. In the example, by watering the root of the tree the watering of the branches and leaves takes place automatically. When the pure devotees of Bengal or other places perform a few karmas out of obligation because of pressure from family traditions, this is actually not performance of karmas, and does not give karmic results, because they do not have faith in the worship of devatas. The Lord says:

asraddhaya hutam dattam tapas taptam krtam ca yat |
asad ity ucyate partha na ca tat pretya no iha ||

O son of Prtha, whatever is offered in the fire, whatever is given in charity, whatever is undertaken as austerity, but which is done without faith, is called asat since it bears no result now or in the next life. BG 17.28