I have searched the web for some possible solutions and advice for your problem. Here are some of the results that I found:

• A simple and efficient method for DNA extraction from grapevine cultivars, Vitis species and Ampelopsis by A. Lodhi, G. N. Ye, N. F. Weeden, and B. I. Reisch, Plant Molecular Biology Reporter, vol. 12, no. 1, pp. 6-13, 1994. This paper describes a method for DNA extraction from grapevine and related genera using cetyltrimethylammonium bromide (CTAB) buffer and polyvinylpyrrolidone (PVP). The method is reported to be effective for removing polysaccharides, polyphenols, and other contaminants that interfere with DNA quality and quantity. The method is also suitable for PCR amplification of DNA fragments.
• A rapid and efficient method for isolating high quality DNA from leaves of carnivorous plants from the Drosera genus by M. Tretyakova, A. Romanovskaya, A. Shmakov, and V. Sizykh, BMC Research Notes, vol. 7, no. 1, 926, 2014. This paper presents a method for isolating high quality DNA from leaves of carnivorous plants from the Drosera genus using CTAB buffer and PVP. The method is claimed to be rapid and efficient, as it takes less than 3 h to complete and yields DNA with high purity and integrity. The method is also compatible with PCR amplification of DNA fragments.
• A simple method for isolation of genomic DNA from fresh and dry leaves of Terminalia arjuna (Roxb.) Wight & Arn., a medicinal plant by S. Kaur, R. Ahuja, S. Kumar, R. Chaudhury, and A. Srivastava, Molecular Biotechnology, vol. 40, no. 2, pp. 167-172, 2008. This paper describes a method for isolating genomic DNA from fresh and dry leaves of Terminalia arjuna, a medicinal plant with high levels of secondary metabolites that hamper DNA extraction. The method involves the use of CTAB buffer with PVP and β-mercaptoethanol (β-ME), followed by chloroform-isoamyl alcohol (CIA) extraction and isopropanol precipitation. The method is reported to yield high quality DNA that can be used for PCR amplification of DNA fragments.

Based on these results, it seems that CTAB buffer with PVP is a common solution for extracting DNA from plant materials that have high amounts of pigments and other contaminants that affect DNA quality and quantity. You may want to try this method or modify it according to your needs and see if it works better than the protocol you attached.

Here are the references

: [A simple and efficient method for DNA extraction from grapevine cultivars, Vitis species and Ampelopsis] by A. Lodhi, G. N. Ye, N. F. Weeden, and B. I. Reisch, Plant Molecular Biology Reporter, vol. 12, no. 1, pp. 6-13, 1994.

: [A rapid and efficient method for isolating high quality DNA from leaves of carnivorous plants from the Drosera genus] by M. Tretyakova, A. Romanovskaya, A. Shmakov, and V. Sizykh, BMC Research Notes, vol. 7, no. 1, 926, 2014.

: [A simple method for isolation of genomic DNA from fresh and dry leaves of Terminalia arjuna (Roxb.) Wight & Arn., a medicinal plant] by S. Kaur, R. Ahuja, S. Kumar, R. Chaudhury, and A. Srivastava, Molecular Biotechnology, vol. 40, no. 2, pp. 167-172, 2008.

https://www.frontiersin.org/articles/10.3389/fmicb.2022.956855/full

https://www.frontiersin.org/articles/10.3389/fmicb.2016.01538/full

Annulohypoxylon sp. strain MUS1, an endophytic fungus isolate...

Article Diversity and Taxonomic Distribution of Endophytic Bacterial...

https://www.frontiersin.org/articles/10.3389/fmicb.2016.01538/full

https://doi.org/10.3389/fmicb.2016.01538

https://pubmed.ncbi.nlm.nih.gov/

https://scholar.google.com/

https://www.sciencedirect.com/

Good luck