Change the 2-pole breaker to a Square D Homeline 30A breaker ($9). Make sure that cable is at least 10 AWG (8 or 6 will do).
Then fit a NEMA 6-30 as you intend.
Then replace those two 15A 1-pole breakers with Square D Homeline 15A breakers. If they power one cable with a shared neutral, use a 2-pole breaker.
Those Siemens QP breakers have no business in your panel. Donate those to someone with a Siemens/Murray panel. This isn't brand loyalty, the breakers do not fit properly and will arc.
Good news, your article 220 load calculations are correct
Upon reviewing 220.82 and your panel photo -- your article 220 load calculation seems to be spot-on in terms of what it includes. Note that you do not need to use the 125% continuous load multiplier for Article 220 purposes as the applicable Code text explicitly refers to the nameplate rating of the appliance.
As to the tapping...
Your existing feeder conductor is indeed correctly sized for 100A -- the 83% rules for dwelling unit feeders/services in 310.15(B)(7) means that you can get by with a cable that can carry 83A, and 2AWG aluminum SE is allowed to carry 90A when terminated on 75°C lugs as per 338.10(B)(4)(a) (text from 2017 NEC, 334.80 is the section that governs temperature ratings and allowable ampacity of NM cables):
(4) Installation Methods for Branch Circuits and Feeders.
(a) Interior Installations. In addition to the provisions of
this article, Type SE service-entrance cable used for interior
wiring shall comply with the installation requirements of Part II
of Article 334, excluding 334.80.
For Type SE cable with ungrounded conductor sizes 10 AWG
and smaller, where installed in thermal insulation, the ampacity
shall be in accordance with 60°C (140°F) conductor temperature rating. The maximum conductor temperature rating shall
be permitted to be used for ampacity adjustment and correction purposes, if the final derated ampacity does not exceed
that for a 60°C (140°F) rated conductor.
Since you used mechanical setscrew-type (Polaris™) splices for your tap, you should be fine with the copper-to-aluminum connections as those connectors are Cu7Al or Cu9Al rated in their common incarnations. Furthermore, the 83% rule for the run wires from the meter-main to your unit still holds thanks to 310.15(B)(7) point 3:
(3) In no case shall a feeder for an individual dwelling unit
be required to have an ampacity greater than that specified in 310.15(B)(7)(1) or (2).
Finally, the formal feeder tap rules do not apply here at all since the conductors from the junction box to the new subpanel are of a size which is adequately protected by the feeder breaker (#2 copper wires in conduit or a SE cable can handle 115A) without further consideration. (In other words, they are not tap conductors in the Code sense of the term -- they would need to be too small to handle 100A in order to be considered as such.)
...and the subpanel in the garage
The installation of the receptacles and subpanel in the garage is close to correct, but needs rectification on a couple of points. The good news is that these are not hard fixes, and should be manageable in a day -- nothing major was screwed up here, as you appear to have your neutrals and grounds sorted, and even though this box is full, that's an acceptable concession to make for something special-purpose.
The first issue is that the 50A breaker is a bit large for 8AWG wire going to a receptacle -- it would be fine if that 8AWG was going to say subpanel lugs via a conduit, as you could use the 75°C column in that case, but most receptacles are restricted to 60°C service, so that limits you to using the 60°C column in the ampacity table for circuits feeding receptacles, no matter the type of occupancy. Swapping the 50A breaker for a MP240 or QP240, presuming that's the correct type for what appears from the interior layout to be a Siemens/Murray spa panel, will correct that, while the NEMA 14-50 receptacle can stay as NEC Table 210.21(B)(3) permits 50A receptacles on 40A branch circuits. (It also permits 40A receptacles, but there are no 40A NEMA receptacle configurations.)
The other thing I noticed in your picture is a panoply of utterly redundant grounding wires, at least if the NEC grounding rules govern unchanged in your jurisdiction -- the RMC nipples are an adequate grounding path from the tap box to the subpanel and from the subpanel to the junction boxes in this case (i.e. not a service, and less than 250V to ground), meaning you simply need to use appropriate gauge ground pigtails from the receptacle to the box to finish the grounding path, given that you are using "bump out" covers on your junction boxes. Furthermore, the bonding bushing where the feeder enters the subpanel shouldn't be necessary, either -- that gets rid of the wire from the bonding bushing to the ground bar. In the end upshot, your ground bar will be rather empty, but that is fine given that you are using metal boxes and metal conduit.
Adding the stove is fine too, once you send all the Zinsco hardware to the smelter
It turns out that adding the stove only adds 16A of factored load, maximum (40% of 40A), to your 220.82 calculation, which puts you at 87.3A of computed service load, which is still within what a 100A service can handle -- the electrician who says you absolutely need a larger feeder to handle the additional load is incorrect. However, you might as well get a new meter-main fitted anyway (with the cooperation of the other stakeholders), as you have a serious problem: not only is your existing subpanel a Zinsco, your existing meter main uses Zinsco breakers as well! This renders this entire electrical installation hazardously unserviceable, as a documented failure mode for Zinsco breakers is them not turning the power off when you place the handle in the OFF position.
At this point, what you do is up to you and your fellow stakeholders. A two-socket, 200A unit of whatever specification your utility accepts is not a bad choice for the replacement meter-main as it's one less piece of hardware to swap if future expansion is called for, while I would fit a 42 space minimum, 200 or 225A main lug panel with ground bars to replace the existing subpanel pyromaniac monster lurking in your closet.
Best Answer
Multi-wire branch circuits (MWBCs) must be on a 2-pole breaker, never ever a twin! Measure between the two hots, that must always read 240V. If it reads 0V, you are overloading the neutral.
What you want to do is easy enough, though. Run yourself an 8/3 or 6/3 cable. In the service panel, land it on a 20A 2-pole breaker.
At the other end, bring it into a large box such as a 4-11/16 square deep box at the location where you would want that outlet. Then bring into that box your two 20A subcircuits, on 12 AWG wire. All the whites get wirenutted together, then black #6/8 to one of the #12 blacks, and red to the other black. Continue the #12 cables on to your outlet locations.
If you ever fit the sauna, you'll need to abandon the MWBC. You can't put 15/20A receptacles on a 40A circuit unless you fit a subpanel here. However if you think you might fit a subpanel, run #6 wire. That will let you breaker for 60A, and a 60A-fed subpanel will easily support the sauna and the 20A circuits at once.