American space experience was all of 15 minutes (Sheppard's suborbital Mercury flight on May 5, 1961) when JFK made his speech (May 25, 1961). However, Yuri Gagarin had made his 90 minute orbital flight some weeks earlier Apr 12, 1961), so the total space experience was a little less than 2 hours. Titov wouldn't make his 17 orbit flight until Aug 6, 1961.

You're right that the shorter mission lengths allowed for a faster ops tempo back then. Gemini flights happened about once every 2 months. That allowed the astronauts to get a lot of experience in a short time. The Apollo lunar missions were several months apart even though there was a launch window about every 28 days.

Using current technology, a Mars mission would take about 2 years due to the planetary alignment. Even if you have enough energy to do a non-Hoffman transfer to shorten the flight to and from Mars, there isn't much you can do to shorten the total mission length because the planets won't be in a good alignment for the interplanetary segment. That limits your windows of opportunity for missions considerably.

You can't simply put the spacecraft in Earth orbit for a couple years to fully test everything because the space environmental conditions at Mars are different enough to cause problems. For example, the solar energy is considerably less at Mars than at the Earth, so thermal systems need to be able to handle a wider range of temperatures. This may seem trivial (and it is well known), but inadequate thermal system capabilities could result in propellant lines freezing* or below normal internal temperatures.

A lot of testing could be performed in Earth orbit. Some more testing could occur on longer missions, perhaps to the moon or an NEO. In space serviceability is a requirement simply because things will most definitely break during a long mission. Equipment aging may become an issue, but many electronics systems are likely to die in the first weeks of operation or to last for years.

Space mission testing was built on the lessons learned at great expense from flight testing. You can take some shortcuts in flight testing, but the result is almost always bad (e.g. B-26 Maruader and B-29 during WWII, F-100 in the 1950s).

*Propellant line freezing is a different problem than having your plumbing freeze at home. Water is almost unique in its characteristic expansion as it freezes. A frozen propulsion line in and of itself is unlike to rupture. However, the freezing represents a blockage that prevents the propellant from flowing. There's also the possibiliy of rupture if melting and expansion occurs between two blockages.