The easiest way to understand this for a maths novice is to leave axial tilt to the end, and just think of this using an abstract sphere for as long as possible.
Step 1 - Angle of incidence to power ratio.
If we ignore axial tilt, and my crude mspaint, the amount of energy absorbed from the sun for any point on the planet at any given time looks roughly like this:
When it's hitting square on - it's 100%. If the sun is parallel to the surface, it's 0%, and hitting somewhere in between it's half as a strong. We can define the power ratio of any point as $P$.
$$
P = \cos\theta
$$
Where $\theta$ is the effective latitude, 0 for on the equator, $+\pi\over2$ for the north pole,$-\pi\over2$ for the south pole. $P$ will be 1.0 (100%) at the equator, and 0.0 (0%) at the poles.
Step 2 - day length
How hot it gets also depends on how long the sun is up. You've given total rotation time, but how can we get day length? (once again, ignoring axial tilt for the moment).
Same formula. When the midday sun is perfectly vertical, the day length is 100%, when it's perfectly parallel to the horizon, the day will be 0% (the sun doesn't rise).
$$
D = \cos \theta
$$
Where $\theta$ is the effective latitude, 0 for on the equator, $+\pi\over2$ for the north pole,$-\pi\over2$ for the south pole. $P$ will be 1.0 (100% day length) at the equator, and 0.0 (0% day length) at the poles.
Step 3 - Both together
We can take some shortcuts here because we're not calculating actual values, only the proportion of the maximum sun energy transfer. Doing the calculus in full adds a scale and constant factor here, but we then remove it to keep within the range 0 to 1. So defining total sun contribution $S$ as the unsatisfying simple $S = D \cdot P$
Step 4 - Latitude and date to effective latitude
Now we start considering axial tilt. 28 degrees is 0.488 radians.
$$
\theta = L + 0.488 \sin ({Y \cdot 2\pi})
$$
Where $L$ is the actual latitude, so 0 for the true equator, $\pi\over2$ for the true north pole, $-\pi\over2$ for the true south pole, and Y is the date of the year relative to the equinox. So 0.0 and 0.5 for the equinoxes, 0.25 and 0.75 for the solstices.
If $\theta$ is greater than $\pi\over2$ (or less than $-\pi\over2$) then the sun is below the horizon all day (ie winter at poles), so if you see this state, it's pitch black 24/7.
Step 5 - Putting it all together
Lat Solstice1 Power Solstice2 Power Equinox Power
-75 46% 0% (dark) 6.6%
-50 85% 4.3% 41%
-25 99% 36% 82%
-10 90% 62% 96%
0 78% 78% 100%
10 62% 90% 96%
25 36% 99% 82%
50 4.3% 85% 41%
75 0 (dark) 46% 6.5%
Step 6 - To temperature.
This is beyond what I can answer, this maths says that the poles in winter should be 0 Kelvin (-273 Degrees C), but we know that's not true, it's a property of residual heat and greenhouse effect and convection through oceans and things like that, which you said to ignore.
But, since you've calculated the average temperature, we can use some trig and calculate that the average power input for the entire sphere occurs on the equinox at the 45 degree latitude band, it's, helpfully, 0.5 at that exact point in time.
This creates a very nice range, where the planet is coldest when the power is 0, it's the average temperature where the power is 0.5, and the hottest when the power is 1.0.
This seems too simple to be true, but the values are within a few degrees of earths so we can use it for reference to be sure we're on the right track.
(Earths orbit is eccentric by just under 2%, slightly favouring a hotter northern hemisphere and a colder southern hemisphere, but the figures should be close enough.)
Your poles are a bit below the average planet temperature in middle of summer. Earths average temperature is currently 14 degrees C. North pole has hit 13 degrees. (South pole has -12 record, but global warming is effecting the arctic more than Antarctica). I'd say this is a loose fit.
We'd also expect peak December temperatures at about 28 degrees latitude south. Hottest city on Earth in December is Brisbane (27 degrees latitude). Peak July temperatures would be at about 28 degrees north. Las vegas (at 36N) is the record holder, Death Valey is also at 36N. I'd say this is also a loose fit.
