You forget the time schedule. Nothing stayed the same along the duration of the civil war, neither quality of troops, morale, meddling, or politics.
Desertion was plaguing every side of this civil war, and surged everytime one side suffered a defeat.
And opposition to conscription and war was opposed on both sides by the peasants who were essentially reduced during the entire civil war to make either a bad choice or a worst one. Cease fire was never a serious consideration especially as the Bolshevils were adamant about keeping their monopoly on power, an objective they were consistent with during the entire conflict ever since they dissolved the constituent assembly. Any time they sought to reach a deal with other socialist factions, it was to join their side on their term, not a true alliance nor a power sharing agreement.
Then the people's army and its cadre turned right after Kolchak's coup against the provisional government in november 1918, coup which alienated the SR support of the anti-Bolshevik government.
Actually, involvement of the Entente powers into White politics was relatively limited and late. Actually, the first to intervene were ironically the Germans who supplied the Volunteers Army with weapons and ammunitions (the White generals disliked the Germans but hadn't much choice) and sent forces into Finland to help the Finnish Whites.
Entente intervention would only really grow in importance beginning with the Japanese occupation of Vladivostok in the late summer of 1918.
Up so far, the Czechoslovak legion hadn't any political agenda and were content with allying to the anti Bolshevik governments propping up under their umbrella. These were set up by SRs, the largest, most significan/powerful opposition force to the Bolsheviks as showed by their performance in the Constituent Assembly.
When the Entente started meddling in and support more right wing leaders and faction, going as far as couping the SR government through Kolchak, they incidentally took away any degree of popular legitimacy the White government was holding. Now it wasn't a government dedicated to reforms, it was made up of conservatives and reactionaries whose displayed intent was to restore the hated pre war socio-economical systems, the privileges of land owners. At the same time, the coup discredited the SRs as they were showed unable to deliver on their promises and cause their own base to switch to the lesser evil that were the communists. At this point, only the most anti communist elements of the SRs stayed on the White side, but had lost all influence.
As of the Red Army being superior to the Czechoslovaks, it wasn't much of a reality early on.
The Red Army was initially more of a militia built on the ashes of the Russian army and wasn't able to fight the civil war at its beginning. It was the army that was chased by the Germans before Brest-Litovsk (fleeing would be the right word), routed and expelled out of the Caucasus by the very numerically inferior Volunteers Army through spring and summer and was forced to abandon practically all of its equipment in the process. It was this makeshift army that attempted to disarm the largest professional military force in Russia in the spring of 1918 and was compelled to flee all the way to Kazan through the summer, from a rail station to another.
The mood of the summer of 1918 was for Bolsheviks one of disasters after disasters, not to mention the left SR coup attempt in July and the assassination attempt against Lenin the month after. Desertion rates were appalling and the Red Army had serious troubles to raise enough men to keep the fighting on.
It was left to Trotsky to transform it into a real fighting force, which he did at great cost, hiring former Czarist officers, enacting conscription and enforcing harsh discipline, which wasn't to give results until the Kazan campaign. That success of the Red Army, still a very close matter, owed as much to Trotsky's capacities as to the overextended lines of the Czechoslovaks and Komuch army. That success was capital for the Bolsheviks as it gave a new breath to the Red Army and a much needed morale boost to go on a counter-offensive. Into 1919, the Red Army wasn't still a high quality fighting force, but none of the armies fighting that civil war were either; at least it was a force capable of standing on the field. Meanwhile, the numerical relevance of the Czechoslovaks faded away as each side was raising its own forces ("quantity has a quality all its own" as per the old say).
Then, into 1919, the advantage was the Bolsheviks', as they had alone an industrial capacity while the Whites relied on foreign supplies, and had a morale advantage as I spoke of above, being the "lesser of the two evils" in that war, so they had more popular support. But this "lesser of two evils" choice wouldn't be without consequences as it would give way to third party insurgencies, the so called Green armies (Makhno's anarchists, Tambov rebellion).