While I don't have information on dates, or any real knowledge of whether these roads were actually signed as US 101, there definitely is a long history of using Mission corridor roads as the gateway toward the peninsula and San Jose.
Historically, there were two main routings between SF and SJ for both cars and trains: Mission routing and Bayshore routing. The Mission routing is older, and it was an easier routing because Mission followed the valley of hills and took the path of least resistance. The routing via Bayshore is more direct, and eventually became the main way of connecting the two cities. Today's BART routing via Daly City to SFO and Millbrae largely follows the old SPRR routing. The modern route of Caltrain follows the newer Bayshore routing, using tunnels to get through some of the biggest hills.
Mission Street, from Daly City, all the way to Downtown, was the main way from San Jose to SF since stagecoach days. When there was a need to route traffic from SJ-SF-Marin (via the Hyde St ferry), northbound traffic left Mission by taking Valencia north, whereas downtown bound traffic continued on Mission.
As far as roads go, cahighways.org has the following information:
Here are some specifics on the routings:
Up to 1932, US 101 from San Jose to San Francisco was US 101W, following El Camino Real, Mission St., Valencia St, Market St. (where it met US 101E coming from Oakland, via a Ferry), Van Ness Av, Bay St., and Hyde St., to the Sausalito Ferry. In July 1932, the US 101W/US 101 E split occurred.
In 1933, US 101W changed at Daly City to the route of San Jose Ave., Alemany Blvd., Bayshore Blvd., Potrero Ave., 10th St., Fell St., Van Ness Ave., Bay St. and Hyde St. to the Sausalito Ferry.
In 1936, US 101W became US 101.
In 1938, the Bayshore Blvd from San Francisco to San Jose became US 101, and Alemany Bl - San Jose Ave - El Camino Real because US 101A. From Bayshore Blvd & Alemany, US 101 continued on Bayshore Blvd., Potrero Ave., 10th St., Fell St., Van Ness Ave., Lombard St., and Richardson Blvd. to the Golden Gate Bridge. There evidently was a lot of infighting as to whether the El Camino or Bayshore would be US 101; some of this is illustrated by the Trees for El Camino Project
In 1940, Alemany Blvd., San Jose Ave., and El Camino Real became US 101, while Bayshore Blvd. became Bypass US 101. The old US 101 and Bypass 101 rejoined in southern San Francisco. The route was LRN 68 for a short time.
EDITED TO ADD:
Now when the San Jose-Alemany route became US 101 in 1933, I would imagine that a lot of traffic would still take the old routing (more or less). Instead of Mission-Valencia, I would imagine that a lot of traffic took the wider SanJose-Guerrero to connect to Market Street. (Hence the name San Jose Ave, showing the preferred way to San Jose instead of Mission.) There was probably no official signage of this routing in any capacity, as the main signage routed traffic along Alemany-Bayshore.
Of course, historically, traffic from Mission to the Hyde Street Pier had to follow some connection to Market (via Valencia, Guerrero or even Dolores) as there was no way for the Mission traffic to go directly to Van Ness because Van Ness did not always extend south of Market. (South Van Ness is newer than most streets in the area, it cuts through the grid and connects to the original routing of Howard Street.)
Van Ness historically, and in the present day, was one of the widest N/S streets in SF, so it has long been a natural routing to the Hyde Street Pier. (Van Ness' width stopped the spread of the fires that burned most of the city following the earthquake of 1908. After the earthquake, it became a major business district as the old part of the city was being rebuilt.