From Special Studies, I extracted the genus who are associated with various symptoms/conditions and then aggregated into a single table below. See Special Studies Updated and A new specialized selection of suggestions links for background.
UPDATE: Just added Possible help for the root of all evil in the microbiome?

Conceptually, these are the ones that will lead to a variety of symptoms and conditions. Two of the more significant ones are available as probiotics:

• Clostridium (i.e. miyarisan (jp) / miyarisan)
• Bifidobacterium (many brands, I tend to favor Custom Probiotics offerings with HMO)
• Prevotella may be available as a retail probiotic soon (see Prevotella copri as Next Generation Probiotics). There is also a US Patent for  Prevotella histocola which implies work on producing it.
• Veillonella probiotics appear to be available in a mixture (ULTRA-VEILLONELLA) however, I have reservations about the seller (site is new – 2022, no species information, no safety or other studies (legally required in the USA), no information about the company, etc. in other words has an unpleasant smell)
• Note that Lactobacillus is far down this list.

CLICK ON THE BACTERIA BELOW TO SEE WHAT INCREASES OR DECREASES EACH

Bacteria Rank	Shifted	Conditions	Sum Of Conditions
Prevotella	Low	22	145
Veillonella	Low	21	113
Clostridium	Low	21	102
Bifidobacterium	Low	20	151
Anaerococcus	Low	17	79
Phocaeicola	Low	17	78
Streptococcus	Low	15	118
Bacteroides	Low	14	72
Peptoniphilus	Low	14	69
Blautia	Low	14	64
Lactobacillus	Low	14	51
Megasphaera	Low	13	51
Erysipelatoclostridium	Low	13	51
Schaalia	Low	13	42
Lactiplantibacillus	Low	13	39
Escherichia	Low	12	36
Porphyromonas	Low	11	46
Corynebacterium	Low	8	39
Fusobacterium	Low	8	31
Slackia	Low	8	27
Campylobacter	Low	7	23
Succinivibrio	Low	6	22
Collinsella	Low	6	22
Roseburia	Low	5	17
Actinobacillus	Low	5	17
Mediterraneibacter	Low	5	16
Megamonas	Low	5	15
Hungatella	Low	5	15
Catenibacterium	Low	4	20
Desulfovibrio	Low	4	19
Haemophilus	Low	4	18